I don’t think I need to spend much time on infinity. Infinitus est numerus stultorum. It suffices to point out that you cannot show me infinity of anything whatsoever. Since everything is finite, including every number, putting them all together will still not get you to infinity. According to math (and also its feisty sidekick, the English language), the number before infinity would be known as the “penultimate” in the series of all numbers. So in my opinion, the last number in the number line is the penultimate.
There is also a convenient common sense method for refuting infinity. If there were infinite numbers, then the Universe couldn’t fit them all in. But clearly the Universe does fit them all in, by the transitive property. It fits our brains, and our brains fit all the numbers. Please see William Lane Craig on this point, for further discussion.
Science and Math Defeated 
Wait, so you are arguing that a number occupies a non-zero amount of space?
Can you show me where the universe manages to fit in pi?
There is no penultimate number. Every number is smaller than that number plus 1.
Mathematically, the number of integers is infinite, i.e. non-finite. i.e. there is no number which represents the number of integers.
Furthermore, in mathematics infinity is not a real number. (meaning it does not belong to the set of all real numbers).
Being generous, you argument amounts at most to a restatement of this mathematical fact.
Is i a number?
“Can you show me where the universe manages to fit in pi?”
I never said pi was infinite.
“There is no penultimate number. Every number is smaller than that number plus 1.”
Easy to say, harder to prove.
“Furthermore, in mathematics infinity is not a real number. (meaning it does not belong to the set of all real numbers).”
Exactly my point.
“Is i a number?”
No. Stay tuned, because I will soon post on imaginary numbers!
>> “There is no penultimate number. Every number is smaller than that >> number plus 1.”
>>
>> Easy to say, harder to prove.
Easy to prove:
Assume there exist an integer number A such that all numbers B A.
This is a contradiction.
Therefore, there does not exist an integer A such that all integers B >”I i a number?”
>> No
Seriously?
i is just about the most absurdly useful number ever invented.
How can e^(pi*i)=-1 if i doesn’t exist?
is -1 a number?
is 0 a number?
(In other words, numbers aren’t real things in the first place, numbers are just ideas. Any idea which has been though exist by definition. Ideas can be useless, and can produce logical inconsistence and contradiction, but they can’t exist or not exist in the ordinary usage of the word. Hence why your argument that the universe ‘can’t contain infinity’ is flawed.)
whoopse! Darn you HTML MARKUP!
Easy to prove:
Assume there exist an integer number A such that all numbers B are less than A.
Let B equal A+1.
It follows immediately that B is greater than A.
This is a contradiction.
Therefore there does not exist a largest integer.
And it follows very easily that the number which represents the number of integers (or cardinality of the set of integers if you will) is not itself an integer.
(check out Wikipedia to find out about formal proofs: http://en.wikipedia.org/wiki/Proof_by_contradiction)
“I will soon post on imaginary numbers!”
And you will soon be ignored on that topic as well.
“It suffices to point out that you cannot show me infinity of anything whatsoever.”
ok…
“So in my opinion, the last number in the number line is the penultimate.”
ok, show me “the last number on the number line of anything whatsoever”
oops.
I don’t know who you’re quoting there, but the point isn’t that I could show you the last number on the number line, the point is that it is logically possible that I could show you penultimate of something. Namely – it could happen in at least one possible world. Ever heard of modal logic? Didn’t think so.
Dearest WTF? Says:
Your whole proof lies on what you are arguing. This is called “Begging the question” and is considered a logical fallacy. http://en.wikipedia.org/wiki/Begging_the_question
You say:
“Assume there exist an integer number A such that all numbers B are less than A.
Let B equal A+1.” Clearly, this step only works if you assume what you’re saying. If we look at notedscholars correct, and highly impressive, thesis we would interject, “Nope, we can’t let B equal A+1 as A is the last number.” Do you understand notedscholar’s idea? ::curious::
Dearest Billion666:
I’m guessing you won’t ignore it based on the FACT that you have now commented on this post and the previous post. Try to bring something more constructive to the debate next time, please.
mrgoodscience.
I’m sorry, but no.
It is a formal mathmatical proof and is correct.
There is never a case where you can’t say Let B equal A+1.
That is the point.
I admit it kinda seems like a begging the question, but because addition is defined for all integers, and A is an integer, it is by definition allowed.
If B cannot equal A+1 then addition is not defined for all integers, and we have a contradiction at the very foundations of algebra.
Think of it this way: Let B=A; Then B+(1-1)=A. However, it would not be the case that (B+1)-1=A, since we could not evaluate the operation in parenthesis, and the associative operation of addition is violated.
Sounds like you have circular logic in your axioms there, urdumb. First of all, nobody ever said the penultimate (P) was an integer. That would be silly. The point is that it’s a real number, as opposed to a fake number.
Now, it doesn’t make sense to add to P, since it’s the last number. It’s like talking about a point outside a topological space. So the axiom commonly used — that addition is defined for all reals — has an obvious implicit corollary: that addition cannot extend beyond P.
This leads to an interesting theorem: P is unknowable
Proof:
Suppose P is knowable.
Then let B = P. Since B \in R, let A = B – 1 \in R.
Since A \in R and A != P, A + C is defined for all C \in R.
Then let C = 2 and D = A + C, and
D = A + C
= A + 2
= (B – 1) + 2
= B + 1
= P + 1
But P + 1 is not defined, so this generates a contradiction. Therefore our original choice of B was incorrect. Since this proof stands for any B, P is unknowable. QED.
Dearest Memethief:
Incredible, and Perfect! Well done!
First I’ve presented a formal mathematical proof, so I actually CAN’T be wrong. Your inability to follow the logically certain argument I have provided for you is not my fault.
So if you disagree with me, no mater how smart you are (you could be like NEWTON+EINSTEIN to the HAWKING SMART!), you actually CAN’T be right. Such is the power of mathematics.
Try this argument about cardinality on for size:
Assume that the penultimate number exist and is real number called P. Let p be the largest integer smaller than the penultimate number. (Clearly 0<=P-p .01
1 -> .11
2 -> .21
…
10 -> .101
11 -> .111
12 -> .121
…
100 -> .1001
900 -> .9001
so there are AT LEST p real numbers between 0 and 1, and AT LEAST p real numbers between 1 and 2 etc…
This means that there are at least p squared real numbers between 0 and p!
In short, if you chose an arbitrarily large p, there are EXPONENTIALLY lager numbers implicitly defined. This is true even if we don’t know what p is. For any value that p could possibly have this will be true.
BUT WAIT! IT GETS EVEN WORSE!
I did said that there were more real numbers defined on an ARBITRARILY arbitrarily small interval!
Have a nice day 🙂
Whoops, cut out the central argument.
First I’ve presented a formal mathematical proof, so I actually CAN’T be wrong. Your inability to follow the logically certain argument I have provided for you is not my fault.
So if you disagree with me, no mater how smart you are (you could be like NEWTON+EINSTEIN to the HAWKING SMART!), you actually CAN’T be right. Such is the power of mathematics.
Try this argument about cardinality on for size:
Assume that the penultimate number exist and is real number called P. Let p be the largest integer smaller than the penultimate number. (Clearly 0<=P-p .01
1 -> .11
2 -> .21
…
10 -> .101
11 -> .111
12 -> .121
…
100 -> .1001
900 -> .9001
so there are AT LEST p real numbers between 0 and 1, and AT LEAST p real numbers between 1 and 2 etc…
This means that there are at least p squared real numbers between 0 and p!
In short, if you chose an arbitrarily large p, there are EXPONENTIALLY lager numbers implicitly defined. This is true even if we don’t know what p is. For any value that p could possibly have this will be true.
BUT WAIT! IT GETS EVEN WORSE!
I did said that there were more real numbers defined on an ARBITRARILY arbitrarily small interval!
Have a nice day 🙂
HTML MARK-UP I HATE YOU!
Assume that the penultimate number exist and is real number called P. Let p be the largest integer smaller than the penultimate number.
Now clearly there are p non-negative integers. But, there are (2*p) numbers between (-p and p). So if we try to count the number of integers on the number line between -p and p, then we have an integer twice as large as the biggest integer, and very nearly twice as large as the biggest number! OH MY!
But wait, it gets worse! There are more real numbers between 0 and 1 then p! To see that this is the case, we only need to realize that every integer number can be encoded as a unique real number on an arbitrarily small interval. Every integer number has a unique sequence of digits. By merely placing a tiny dot on the left side of that sequence, and a 1 on the right we generate a unique mapping of every integer number onto the (non inclusive) interval between 0 and 1!
Examples:
0 -> .01
1 -> .11
2 -> .21
…
10 -> .101
11 -> .111
12 -> .121
…
100 -> .1001
900 -> .9001
so there are AT LEST p real numbers between 0 and 1, and AT LEAST p real numbers between 1 and 2 etc…
This means that there are at least p squared real numbers between 0 and p!
In short, if you chose an arbitrarily large p, there are EXPONENTIALLY lager numbers implicitly defined. This is true even if we don’t know what p is. For any value that p could possibly have this will be true.
BUT WAIT! IT GETS EVEN WORSE!
I did said that there were more real numbers defined on an ARBITRARILY arbitrarily small interval!
Have a nice day 🙂
urdumb, you haven’t proven anything about P. All you know by the end of your argument is that 2*p \le P and p*2 \le P. This is not a contradiction by any leap. If you do find that p*2 \gt P, then you have chosen the wrong P (see theorem above, re unknowability of P)
“First I’ve presented a formal mathematical proof, so I actually CAN’T be wrong. Your inability to follow the logically certain argument I have provided for you is not my fault.”
I presented a better-formed mathematical proof than yours. Your disorganized proofs are obviously indicative of axiomatic flaws.
(and I feel your pain; it’s irritating trying to do math and HTML at the same time 🙂
memethief: The point is, it is true for any P you choose.
So, if P is a real number, then there is a real number larger than P. If you cannot see that is the case, I cannot help you. I have proved it formally, and given you several ways to see that there by making the largest number a real number you implicitly define larger real numbers.
You proof of the ‘unknowability’ of P, is exactly equivalent to mine, (except introducing some extraneous steps) and, what is worse, proves that any number you chose for P cannot be P! So you proved that if you assume that any number is P, then that number is not P!
How can you not see that this means P does not exist?
And by the way, since p is at most one less than P, as long as the largest real number is greater than 1, then 2*p is greater than P. additionally I showed you that by defining P you implicitly define P*2, and P to the P.
(that is you if you count the number of real numbers you implicitly define by choosing P, that number must always be greater than P.)
“The point is, it is true for any P you choose. ”
Um, no, that’s the opposite of the actual result. It’s /false/ for any P you choose. That’s the whole point you don’t seem to be grasping.
“You proof of the ‘unknowability’ of P […] proves that any number you chose for P cannot be P! So you proved that if you assume that any number is P, then that number is not P!”
Exactly — you’re finally getting the point. Any assumption you make about the value of P is proven to be false. It’s all very quantum.
“How can you not see that this means P does not exist?”
Wait, what? And here I thought you were starting to understand hypofinite math. I even laid out my proof step by step so you could follow along (the “extraneous steps” you mentioned).
Okay, here’s the crux of things: if you set a variable A to P, and with it find a number greater than P, then it is *obvious* that you didn’t have P in the first place, since by definition there is no real number greater than P. Therefore A does not exist, by the transitive property, as proven above.
And by the way, you need to work on your proof style. An ideal *formal* mathematical proof is succinct and to the point — hence my 12-line proof of the hypofinite unknowability theorem. As a rule of thumb, phrases such as “OH MY!” and “But wait, it gets worse!” don’t belong in a proof. In fact, I would suggest leaving exclamation points out entirely unless you’re using factorials.
“that is you if you count the number of real numbers you implicitly define by choosing P, that number must always be greater than P.”
Right, so you cannot choose any P. The set of reals is uncountable large, by the way, but the cardinality of any countable subset of R is less than or equal to P.
“As a rule of thumb, phrases such as “OH MY!” and “But wait, it gets worse!” don’t belong in a proof.”
Yes, indeed the post you are referring to was not a proof, but instead a persuasive argument to show the absurdity of your position. I posted several informal arguments to demonstrate that important contradiction are created by asserting that there exist a real number larger than all other real numbers.
“Exactly — you’re finally getting the point. Any assumption you make about the value of P is proven to be false. It’s all very quantum.”
First, not quite, only asserting that P is a real number is false. You may be able to define P if you make a new algebraic set to define reasonable operations on it (and of course people already have).
Secondly, If you wish to assert that a number which cannot in theory or practice be assigned any real number value exist and has a real number value that is fine.
Personally I would argue that if any number you can choose is clearly not P, then no number CAN BE P, then no number IS P, and P IS NOT a number.
I choose to say such a number is nonexistent, but if you wish to assert self contradictory concepts exist by some semantic game, I really couldn’t care less.
BTW. quantum refers to a discrete system, it’s not some wibly wobly adjective to discribe vacuous concepts invented by the mathematically inept.
At any rate, I have done my part and should be able to follow my arguments if you choose.
It is clearly impossible to force someone to adapt any proposition, and I am glad that this is so.
Finally I need to inform you that you are mentally ill. This may be a benign health issue (I hope it is!), in which case I advise merely mindful self observation. If you notice any other symptoms of paranoid delusional behavior in yourself you should seek immediate professional help.
-Peace out brother.
Urdumb:
“At any rate, I have done my part and should be able to follow my arguments if you choose.”
I don’t know who “memethief” is, but I doubt very much he/she can choose for you to follow your own arguments. That’s up to you, and I doubt very much you can do it.
Second, you say to this “memethief”:
“Finally I need to inform you that you are mentally ill. This may be a benign health issue (I hope it is!), in which case I advise merely mindful self observation. If you notice any other symptoms of paranoid delusional behavior in yourself you should seek immediate professional help.”
Did you get that line from a how-to book on Stalinism?
http://mentalhopenews.blogspot.com/2007/07/activist-sent-to-mental-clinic-russian.html
Sorry, you can’t just diagnose people who disagree with you.
……that is, unless you are Joseph Stalin.
“Finally I need to inform you that you are mentally ill.”
It is unfortunate that novel hypofinite mathematical concepts prompt such ad hominem attacks from closed-minded traditionalists. But I hope that eventually you will come to understand these advanced ideas.
Until then, take care sister.
urdumb Says: “The point is, it is true for any P you choose.”
Please use the indefinite article for any “points” being made, so as not to falsely assert we all share the same point.
Note that I don’t yet know where I stand on this issue.
But *a* point is, you cannot chose the penultimate – it simply *IS* penultimate.
It chooses you, you could say.
But let’s not get into the whole infra/supra/sublapsarian debate…
Pardon me, but it seems to me that your debate is lacking some fundamental definitions. Could you please define what you mean by “number”?
Also, it seems to me that your argument about numbers not “fitting in” the universe depends on the assumption that only “actual” objects (note: I quote this to signify the need for a discussion) are permitted for use in discourse. Does this mean we cannot talk about things that do not exist? For instance, can we discuss unicorns and Odysseus on reasonable logical grounds? Can we discuss the number one? What is the number one?
I believe that if you provide an axiomatic system that permits the natural numbers [e.g. Peano arithmetic] then we will find that either your penultimate number P does not exist, or P = omega where omega is the first infinite ordinal number, defined as the first non-zero number with no successor.
Nevertheless, I think all of your claims require more foundational support in order to be logically sound.
Dear Notedscholar,
If your “countable numbers” are defined to be “quantities”, then do they include numbers such as a googol [10^100] or a googolplex [10^googol], the latter of which well exceeds the number of particles in the universe, the size of the universe in cubic Planck lengths, the time since the beginning of the universe in Planck times, and indeed the product of all three of these numbers. Since the googolplex is far larger than any quantity in the universe, is it a number? Furthermore, since psychologists would agree that no human has an innate intuition for numbers this large, would it really be fair to say that such large numbers are “built into our linguistic-cognative makeup”?
Also, since you have stated that P+1 and P-1 are not defined, we can only conclude that P is nonzero and the successor of no other number (in the Peano sense) and therefore must be an infinite ordinal.
I believe that the root of your problem with infinity is a misunderstanding of the proof that there are an infinite number of natural numbers.
The proof proceeds thus (using the standard Peano axioms for the natural numbers starting at 0):
By the Peano axioms, every natural number n has a successor that we will denote n + 1.
Consider the set N containing all the natural numbers 0, 1, 2, …. Assume this set is finite. Then, N = { 0, 1, 2, … , P } for some largest natural number P. But then P has successor P + 1, so the question is: is P + 1 in N?
Well, for any natural number n in N, we find that n < P because for P – n successors after n is P. That is, n + (P-n) = P.
However, P+1 is not strictly less than P because P+1 is the successor of P, so we conclude that P + 1 is not in N. This contradicts our assumption that P is a natural number (because all natural numbers have successors that are natural numbers) and so we conclude (by contradiction) that no such P can exist.
Dear Frege,
First off – I’ve read about Frege, you sir are no Frege.
The concept of “number” is an irreducibly simple concept – it is an assent-compelling notion built into our linguistic-cognitive makeup. Put simply I would say that a number is a quantity, but this is necessarily vague for a properly basic belief.
You ask if we can “talk about” things that do not exist. We (almost) certainly can. Barring some kind of Berklean Idealism, there is no metaphysical problem in doing so – and certainly not the ludicrous metaphysical problem you suggest.
In answer to another one of your inane questions – yes, we can “discuss” the number one. We can even “discuss” the number “infinity.” I don’t see what the problem is here.
Ex Hypothioso, P *cannot be* the first infinite ordinal number. And there is no internal contradiction here.
It is possible that my claims require “more foundational support.” I try to acknowledge this by including “part one” in titles where I believe more discussion will be fruitful. For example, I have not yet developed an axiomatic system. But I will soon in the context of what I have called “Countable Numbers.” Hold on to your horses, lady!
Pingback: notedscholar- for real? « Epsilonica
The existence of P implies the existence of infinity.
Suppose you have a Turing machine–a computer program, in other words–that is programmed to take any natural number written in binary (0 or 1) and return its successor. Its operation can be described as follows:
1. Start at the digit with the smallest place value.
2. If the digit is 0–or if there is no digit written–change it to 1 and terminate.
3. If the digit is 1, change it to 0 and proceed to the next digit.
4. Return to step 2
Now, if P+1 is defined, feeding this machine P would return P+1. If it isn’t defined, then the program would not terminate.
Both of these outcomes wreck P. If P+1 is defined, P’s definition is, of course, faulty. As for P+1 not being defined, look at the program logic. What kind of strings would it terminate on? It terminates on any string containing 0, and it terminates on any string with a blank value–a finite string, in other words. If it doesn’t terminate on P, then it’s because P is *infinite* and has no zeroes. So either there is no P, or P is infinite. Either way, you lose.
Pseudononymous Coward:
Thank you for your question. The fact is that the Turing machine in question would never be able to reach the Pth bit. After all, doing so would require approximately 4P operations — an impossible feat. In fact, feeding the program P as input would take more time than the universe has, and would take up more memory than could conceivably be allocated. The same is true even of relatively small numbers such as 10^10^10.
So the universe would die a heat death before the program terminates, which underlies an important fact about the Penultimate — while finite, there is no conceivable way to have P of something.
Hope this helps,
I = infinity
P = Penultimate
F = finite
I = P + 1
X = 1 / 100
Y = X + 1
Z = Y * 100
Z = 101
A = P / 100
B = A + 1
C = B * 100
C > I ???
F = I – 1 correct?
K = P / 100
L = K + 1
M = L * 100
M > F ???
somewhere math is flawed…
Pingback: This weeks finds in General Mathematics « Epsilonica
notedscholar: “I have not yet developed an axiomatic system.”
Then you, sir, are no scholar. Nor do you have any justification for making any claims. I suggest you read up on your basic logic.
By the way, do you even know what a ring is? I suggest you do some reading (Dummit and Foote’s algebra book is a good one that might help you learn some of this). When you do learn what a ring is, think about the integers. (You will hopefully notice that the integers form a ring.) In particular, consider concept of closure. From what I have read of your comments (and your laughable attempt at writing proofs), it seems like you have a lot of trouble understanding closure, so pay close attention when you are learning about this. Good luck!
Again in this post I feel the author misses the point. No one claims infinity is a number. People use the concept of infinity for practical purposes, as they do imaginary numbers, or probability. The author has no point. What does “refuting infinity” even mean? It is a meaningless statement!
Not to interrupt the mathing here, but how has no-one noticed this:
“So in my opinion, the last number in the number line is the penultimate.”
The last of a thing can’t also be the penultimate anything by definition.
Jason,
You ponder,
I do wish the world were as beautiful as what you describe, full of maximally rational agents. But alas!
NS
Dear Bongs,
You vomit,
Yet you give no justification for this primitive intuition. What you are pushing is just a quasi-metaphysical dogma which is convenient for your mathematical schema.
NS
you are more or less correct. but this concept is a little tricky.
X+1
Think up any number for x. I guarantee you will always be able to add 1.
Myrmidon,
This is one of your few comments that rises to a level worthy of my dismissal.
It is true that you will always be able to “add 1” to any number for x. So you would get, for example, “Penultimate + X.” However, while this is a real expression, it is not a rational expression. Remember that I am not talking about purely formal nonsense language.
Cheers,
NS
Hi,
I disbelieve what you say, but I am genuinely curious, and willing to be brought around.
So I was wondering if you could explain your argument for refuting infinity – which seems to hinge on using the “transitive” property to deduce that the concept of inifinity leads necessarily to physical infinity, which is absurd.
My first question then is what is the “Transitive” property, in this context. My second is why can we not have a useful, logical and rigorous concept of infinity, without there necessarily being an actualy, physical and realisable infinity.
I basically believe in the former and not the latter, and see no contradiction in this position.
I look forward to your reply,
Charlie
“Yet you give no justification for this primitive intuition. What you are pushing is just a quasi-metaphysical dogma which is convenient for your mathematical schema.
NS”
Penultimate quite literally means ‘second to last’ and so the last thing and the penultimate thing cannot be the same thing in the same set. Again, quite by definition.
Infinity is not a number. Infinity is a concept…or something.
Just to cut an end to this nonsense.
The set of real numbers is defined as a Dedekind-complete ordered field. (It can be shown that there exists exactly one such field.)
Dedekind-completeness implies the so-called Archimedean property:
“For any real number R, there exists a natural number N so that N > R.”
Since natural numbers are reals as well, this implies:
“For any real number P, there exists a real number R so that R > P.”
Which is the logical inverse of:
“There exists a real number P so that for any real number R, R ≤ P.”
So, P is not a member of the set of real numbers. (In fact, the proposed definition for P is exactly how positive infinity – a surreal number, NOT a real one – is usually defined in mainstream mathematics.)
Merely stating that P exists, and is a real number (that is, a member of the Dedekind-complete ordered field known as “the set of real numbers”) does not make it so – it must be proven first.
Someone proposed the concept of “knowability”. However, the proposed “set of all knowable numbers” is known in mainstream mathematics as simply “the set of real numbers”. By contrast, the “set of all real numbers, knowable or otherwise (including P)” is commonly called “the set of surreal numbers”.
Also, “the number before infinity would be known as the “penultimate” in the series of all numbers”. Problem is, Dedekind-completeness also implies that between any two real numbers, there exists a third real number – therefore, there is no number “right before” another number, there are always more numbers in-between.
I agree with Noted Scholar, but i believe that he does not go far enough. Not only is ifinity an unshowable, but so is any mathematical idea. Grind the universe down to the smallest bit, and show my an atom of three, a proton of addition, or a quark of division.
Please note that i do not believe that they do not exist, i only believe that they can’t be shown. This arguement can also be applied to mercy, honesty, and justice, all of which i believe in.
What is the numerical value of P?
Also, a friend at my first college by the name of Edward Norris expressed a similar viewpoint.
The argument that there are a finite amount of numbers is pure rubbish. A proof:
A. Assume the contrary; there is a highest number P defined that no number is greater than P.
B. Let p = [P], i.e. the largest integer P.
I. But, by definition, nothing can be greater than P.
J. From H and I, P is a self-refuting concept.
K. From J, there is no highest number, and ergo there are an infinite number of numbers. Q.E.D.
I can hear a few objections around the corner:
“You can’t add to the penultimate, by definition.” Basic mathematics states you can add any two finite quantities. Since the penultimate, by definition, is finite, it can be added to other numbers. There is no way around this without contradiction or special pleading.
“Zero doesn’t exist, so E and all that follow are invalid.” How many flying mothers-in-law do I have, then?
“Your argument is bunk, because there aren’t infinite numbers.” Special pleading and argument by assertion, and nothing more.
So, there are an infinite amount of numbers, and this follows logically from the existence of whole numbers. Any arguments otherwise are rooted in contradiction and/or special pleading. I win.
Mr. or Mrs. Rex,
Your pseuoname is as old – and outlived – and extinct – as your ideas.
“The set of real numbers is defined as a Dedekind-complete ordered field.”
This is your first mistake. Sets are useful fictions, but it doesn’t follow from their formal representations that numbers have any particular property.
“For any real number R, there exists a natural number N so that N > R.”
This is true for any particular number. But there is no number named “infinity.”
“So, P is not a member of the set of real numbers.”
Your proof is clever, but as the saying goes, the clever cake has no frosting. Let’s be clear here that I do not require that P be a “member” of some “set” fiction that you formally define. The Axiom of Choice in non-standard set theory says that all sets are possible. From this it follows that all properties of sets are possible, from which it follows that you can construct a set without P.
“the proposed “set of all knowable numbers” is known in mainstream mathematics as simply “the set of real numbers”.”
I don’t think mathematics makes pronouncements in epistemology. In any case, mathematicians claim to know all sorts of things about non-real numbers, e.g. hyperreal numbers.
“Dedekind-completeness also implies that between any two real numbers, there exists a third real number – therefore, there is no number “right before” another number, there are always more numbers in-between.”
This presupposes the coherence of an infinite number of rational (or irrational) numbers, but that begs the question.
Un Noticed,
First, I detect in your pseudoname some humility; don’t worry, sometimes recognition comes later. As the saying goes, “Einstein wasn’t famous in a day; nor was Newton.”
As for your point, I think it beautifully recognizes the absurdity of human existence. Some things we HAVE to believe in, like honesty (if we didn’t believe in honesty, then how could we trust ourselves?). However, I hope that you can be convinced that CERTAIN – not all! – mathematical concepts are NOT like this. Infinity is just one of them. There is nothing necessary about infinity – after all, even folk thinking yields paradox. Not so with the other concepts you mention.
As for subatomic particles, I have some work on that on the blog, if you care to check.
Best and hope to see you again soon,
NS
The author of this article clearly misunderstands the meaning of infinity. In math, an infinite value has been increased without limit.
“The author of this article clearly misunderstands the meaning of infinity. In math, an infinite value has been increased without limit.”
You claim that someone has increased an infinite value without limit. Who has done this? And when and where have they done it?
Not waiting in anticipation because I know there is no good answer or at least no good answer that doesn’t purposely misunderstand the issues,
NS
1st, the person that used infinity is the one that raised it.
2nd, if you have a problem with this definition, please, PROVIDE YOUR OWN, ydn
NS is a dafty ninny,
Just because someone else provides a bogus definition of a non-concept, doesn’t mean I should provide a non-bogus definition of a non-concept. Non-concepts, by definition, have no definition.
Hope this clears up your fog,
NS
That may be true, but if you want to have a rational debate, I need to know what you mean by infinity.
One thing I find interesting about infinity is that the number of natural numbers and the number of rational numbers are equally infinite.
Do you have any equations that back up your hypothesis?
also, 370H55V 0773H
Where does the universe fit in .999? Or does it, like pi, have finitely many digits?
Let’s assume that p, the penultimate number, is the last number on the number line. We will also assume that p is in the set of real numbers. Say someone were to find p on the number line. One before that would be equal to p minus 1. Let’s call this number o. If we add 2 to o, then we have p + 1. Therefore there would be an infinite amount of p, defeating the purpose of the “Penultimate number”.
More to follow.
To disprove that the universe can’t hold infinite numbers. First, let’s assume that the basic rules of math apply (multiplication, division, addition, subtraction). We will also assume that the number .5 (1/2) is in the set of all real numbers. Let’s say that the universe has x particles (a finite number). We will then divide all those particles in half. x /.5
We continue to perform this operation which can be continued on infinitely never reaching zero, thus the universe can hold an infinite amount of things.
Guys. Infinity is a real number. I don’t know what planet you think you’re from but infinity is a number.
I don’t understand this site, quite honestly. It is scientifically proven.
If the penultimate means ‘second to last.’ Therefore, penultimate is NOT the last number.
@titankiller exactly!
if P is penultimate, then p+1. It will always get bigger. There are infinite number of numbers.
Therefore, infinity IS a real number.
Jeeeeeeeeeeeeeessssseeeeeee!
Simple. Pi occupies the entire universe. Your proof is invalid.
Ok. So you don’t think that infinity is a thing. You Think that there are no finite sets. Ok. Let’s take the set of all integers including zero. Let’s say it has length x. Then the integer x must be on the list. In addition to that, x+1 must be on the list because zero is to. For example, if one was the largest number, then there would be 2 items on the list: one and zero. good? Now let the penultimate number be y. The list is y+1 length, and therefore y+1 is a number. QED there is no largest number
The way addition is defined, you can add any real number to any other real number, and get a real answer. The same goes for counting, natural, rational, irrational, algebraic, transendental, imaginary, and complex numbers, as well as integers, quaternions, octonions, and sedenions. This is the closure property of addition, and applies to every general set of numbers I’ve ever worked with. In addition, by definition, given any real number a, it is always the case that a+1 > a. By the closure property of addition, we also know that a+1 is a real number. This means that the can be no real number P such that every real number is less than or equal to P (the definition of the maximum of a set). Similarly, a–1 < a for all real numbers a, and subtraction is also closed for real numbers, so a–1 is a real number. Therefore, there can be no real number -P such that all real numbers are greater than or equal to -P (the definition of the minimum of a set). Since no real number can be the maximum or minimum of the set of all real numbers, the set of all real numbers is infinite. QED
That said, there is a philosophy of math which ignores infinity. I write about in your page about 0.999…. It doesn't mean we can't use it, though.
Also, why do you keep calling your last, finite real number "penultimate"? Penultimate means next-to-last, which implies that P is the second to last number. Ultimate means last.
Finally, I hate when people imply that any number that is not in the set of all real numbers is a "fake" number. There are many "true" numbers that are not in the set of all real numbers, but that doesn't make them fake.
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So, you think the integers are bounded? You seriously think we might run out of them some day, reach a number where you can’t add one anymore? This blog is a joke, right?
I have to point out, infinity isn’t a thing who’s existence is debated. Our mathematical system is built on certain axioms, one of these is that numbers just *keep going* The universe doesn’t have to “fit” them because they don’t occupy physical space. Our brains don’t need to fit them, and indeed can’t, because we dont memorize literally every number, just combinations of the base 10. Also, serious question. Do you genuinely believe the absurdity you spout on the internet which literally would not exist if the thugs you say are false were indeed false, or are you just a troll looking for likes and shares?
Mutedajar, Why would I think that integers are “bounded”? We can count them freely, without constraints, and nothing stops them from moving to the right or to the left. Only TWO numbers – Penultimate, and Zero – can only move in one direction (left and right, respectfully), but even that is not fully bounded, since they are free.
Matt, very funny of you and nice try, but no cigars to be had. You cannot just faithfully and religiously assert the truth of your precious theories and appeal to the authority of “not debated.” You have to address the arguments offered.
Out of curiosity, how many numbers would you say are between one and the penultimate? My math teacher asked me this when I put your argument forward and now I’m a bit confused.
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It’s very simple. Both 0.999… and 1 are the smallest numbers greater than every element of the sequence: 0.9, 0.99, 0.999, … That makes them both be the LUB (least upper bound) of the sequence. Because the reals are well ordered, there can only be one LUB, therefore 0.999… = 1
Another way, if 1 – 0.999… = d > 0 then let N = log10(1/d). Then for every natural n > N, we’d have 0.999…9 (n 9s) > 0.999… (infinite 9s) and that is absurd.
Infinity is not a natural (or real) number. There is no penultimate or ultimate NATURAL number. Infinity is simply about the idea of endlessness. Like with all math, it makes no claims about physical reality at all – that’s for the sciences.
Oh so simple, you say? You say this to me on my journal? Let’s take a look about.
“Both 0.999… and 1 are the smallest numbers greater than every element of the sequence: 0.9, 0.99, 0.999, …”
Um, no. 0.999… is smaller than 1. So only 1 is great than every element in the sequence.
“That makes them both be the LUB (least upper bound) of the sequence.”
You cannot “make” something into what it is not, no matter how hard to try it!!
“Another way, if 1 – 0.999… = d > 0 then let N = log10(1/d). Then for every natural n > N, we’d have 0.999…9 (n 9s) > 0.999… (infinite 9s) and that is absurd.”
Absurdity is always in the eyes of those who behold it. What is more absurd? That 9s go on forever, or that they magically turn into 1?
“Infinity is not a … real … number.”
Finally, something we agree!
“There is no penultimate or ultimate NATURAL number.”
Um, I literally talked about the penultimate. It is real if anything is.
“Infinity is simply about the idea of endlessness. Like with all math, it makes no claims about physical reality at all – that’s for the sciences.”
This is beautiful for poetry, but not very accurate. Did you know that scientists also leave reality? Especially Brian Greene.
Best luck in your continued studies,
NS
As both 0.999… and 1 are greater than every element of the sequence 0.9, 0.99, 0.999, … and as infinitely many terms of that sequence are greater than EVERY number less than 0.999… and 1, your claim that 0.999… < 1 is manifestly false. You merely assert that 0.999… < 1 even though I have proven that that's absurd. Can you tell the difference?
Physical reality is irrelevant. No numbers at all exist in physical reality. No mathematical object of any kind whatsoever exists in physical reality. It is entirely irrelevant that people can think of other ideas that don't have a physical existence. It doesn't contradict what I said, I can't imagine what your point (Brian Green link) was supposed to establish. Was there a point, or is it just you trying to divert from the subject?
Wish away, but I understand this topic and you don't.
notedscholar, you said, “Um, no. 0.999… is smaller than 1. So only 1 is great than every element in the sequence.”
Please indicate even as much as one element of the sequence that is greater than 0.999…
TruthJenny, I want to be the first to apologize that I did not respond to you. Your maths teacher is very wise! To answer, there are zero numbers between one and the Penultimate. This is why it is the Penultimate. (As far as I know, one is itself the Ultimate. This is why THE ONE is so important in many religions. Jordan Peterson, if you are reading this, please put THAT archetype in your modernist pipe and post it.)
Best,
NS
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notedscholar, that means that the penultimate number is 1. Surely 2 is bigger, then 3 etc.
Dear Chris,
You are begging the question against me! Only in your system that says there is infinity will it be that there cannot be zero numbers between 1 and 0.999… .
Best,
NS
In “my” system 0.999… = 1, and there are no numbers between them – not even infinitesimals at any order. It is you that said (to TruthJenny, ” there are zero numbers between one and the Penultimate”. Are you saying that doesn’t mean that the penultimate number is 1, i,e they are different but don’t have a difference or more relevantly a number between them?
You are discussing some other “system” of your own. Have you got a page somewhere with your definition of your number system, or are you merely adding a wannabe axiom to the usual field definition even though it is completely incompatible with it. See https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Synthetic_approach
for an idea of the sort of details that you need. You can’t just add an axiom that contradicts other axioms.
Why?
Not “Why is this true?” but “why bother?”