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2016-03-29
Challenge to the Mathematic and Logic Community
The challenge is very simple: Give a full and explicit proof of the existence of Infinity.There is only one restriction: it is not allowed to refer to "proofs" already known. If you believe Cantor has proved Infinity, you cannot just refer to his work, but have to state explicitly what and how you think he has proved it.
I wish you luck.

2016-03-31
Challenge to the Mathematic and Logic Community
The set of integers.  There is no "last integer".  If L were the last integer then L + 1 would be bigger which is a contradiction.   There is your basic first infinity.  The cardinal number of the integers is infinite.  Technically it is called Aleph 0. 

2016-04-06
Challenge to the Mathematic and Logic Community
Reply to Robert Kolker
I tend to agree with this clear and simple approach. It expresses the general intuition of what is Infinity: whatever you do, you can do it again. I am afraid though that it cannot be considered as a proof of Infinity in the Cantorian sense. What it does prove is that whatever finite integer you may think of, you can always think of an even bigger finite integer. You are still dealing with finite numbers and the existence of an aleph is still a mystery.


2016-04-15
Challenge to the Mathematic and Logic Community
if I see what you mean, the proof of infinity consists in the possibility of iterating an operation. This is bad infinity, according to Hegel: the good proof instead, in his view (see the  note on matthematical infinity added in the second edition of the "Science of Logic") , is given by the fraction 9/7. The difference between 9/7 (good infinity) and 1.285714... (bad) is due to the fact that in the first case you do not have to iterate anything, you already have the infinity "all contained" in the fraction (so Hegel). What do you think of this "proof"?

2016-04-15
Challenge to the Mathematic and Logic Community
Reply to Francis MC d'Agostini  http://philpapers.org/post/14574
[sorry, pressed on Follow up instead of reply]

I'm not sure I am qualified to answer your question. I found The Phenomenology of the Mind already quite a challenge, let alone the Science of Logic. If I remember correctly, Hegel is here pointing at the ("dialectical") relation between Infinity as a state, the fraction, or the famous acorn, and the Becoming of Infinity,  1.285714...., or the growing tree. I don't know if the qualifications of good and bad infinity are really Hegelian. Anyway, we are still a long way from a proof. The main difference between Hegel's view and Mathematical Induction as expressed by Robert Kolker above is I think the language used. Nobody denies the intuition of Infinity, least of of all me. But how can you get from this intuition to a proof that could justify different levels of Infinity, and therefore concepts like Omega or Aleph?
 I personally think it is not possible, that these concepts are metaphysical representations akin to that of God. Because mathematicians seem to believe in their legitimacy, and even dare make calculations based on them, I expect them to be able to give a mathematical proof of the existence of these concepts. As I see it, it would be a miracle if they could, still I have to ask.


2016-05-16
Challenge to the Mathematic and Logic Community
I don't think it is possibly to proof that. F.e. in Z.F. set theory the existence of infinite set is assumed by accepting the axiom of infinity:

\exists X \left [\varnothing \in X \land \forall y (y \in X \Rightarrow S(y) \in X)\right].

i.e. there is such a set X such that for all y if y is a member of X, then the successor of y is a member of X. Since successor function S(x) can be repeted an infinite time, X is infinite ( i.e. S1(S2(S3...(y)...)) ).

2016-05-17
Challenge to the Mathematic and Logic Community
I hope you won't mind if I tell you to read what I wrote above. I am afraid I would be otherwise only repeating myself.
Just this: if Infinity is not provable, what is the value of concepts like the Continuum, different levels of infinity, etc. And therefore, what is the value of Set Theory itself?


2016-06-06
Challenge to the Mathematic and Logic Community
I'm a scientist, thus I know there is no proof of anything, just evidence, least of all infinity.  Proving is delimiting.  It is oxymoronic to think you could delimit infinity.  ;-)

2016-06-07
Challenge to the Mathematic and Logic Community
First, I hope we are not confusing here two different concepts of infinity. There is the "intuitive" one, which is maybe philosophically interesting, but there is also the well-defined one (f.e. Dedekind infinity).
Now I was personally talking about the well-defined one, which is assumed in most of the axiomatic set-theories. This is fine, we can't prove everything in mathematics. But to think that from this follows that set-theory has no value is absurd. The point of any axiomatic theory is to capture all truths of certain "structure", since axiomatic theory is implicit definition of a certain structure. 

2016-06-07
Challenge to the Mathematic and Logic Community
Reply to Tami Williams
Okay, but that is the easy part. Does that mean you think Cantor's conception of infinity (the different levels or alephs) to be nonsense also? I do, but I am really interested in a scientist's point of view.

2016-06-08
Challenge to the Mathematic and Logic Community
Can you be more specific? Why would Cantor's conception of infinity be nonsense?

2016-06-08
Challenge to the Mathematic and Logic Community
I am afraid it is not good enough. I would have no problem with the idea that mathematical infinity is a mere axiom because then its truth value would be irrelevant, only the formal validity of the following argumentation and the theorems making use of the axiom. Such a formalist approach is of course legitimate and needs no justification. But Dedekind and Cantor go beyond such a formalist view. They present arguments for the rationality and validity of their principle of infinity. That make them more than mathematicians in my book. They are presenting a metaphysical system. The question now is, what is the value of this system? Should one consider it as a metaphysical edifice one can agree or disagree with? And that would be my view. Or should the scientific pretensions of this edifice really be taken seriously? I have tried to show that this is not the case, and I would be really interested in objections from the "scientific" side of things.

2016-06-09
Challenge to the Mathematic and Logic Community
Well, I must admit I'm not too familiar with the original ideas of Cantor and Dedekind. But I should note that "Axiom of Infinity" claims that there is an infinite set, hence it's not claiming about infinity.

2016-06-13
Challenge to the Mathematic and Logic Community
Please see:
The Liar Paradox (and other beasties)

Set Theory: Mathematics or Metaphysics?


2016-09-23
Challenge to the Mathematic and Logic Community

Give a full and explicit proof of the existence of Infinity.

The infinity cannot be proven as it is has to be defined so we have to have common understanding what the term means:

From The Axiom of Infinity (1904) by Bertrand Russell:

The first step is to demonstrate that there is such a number as 0. The number of things fulfilling any condition which nothing fulfills is defined to be 0; and it may be shown that there are such conditions. For example, nothing is a proposition which is both true and false. Consequently, the number of things which are propositions that are both true and false is 0. Thus there is such a number as 0.


We next define the number 1 as follows. The number of terms in a class is 1 if there is a term in the class such that, when that term is taken away, the number of terms remaining is 0. That classes having one member exist is not hard to prove; for example, the class of things identical with the number 0 consists of the number 0 alone, and has only one member.


We proceed in like manner to the number 2, and we prove that the class consisting of the numbers 0 and 1 has two members, from which it follows that the number 2 exists.

We first prove the principle of mathematical induction – a principle which, in this domain, does work for us such as could hardly be expected but from an etcetera. This principle states that any property possessed by the number 0, and possessed by n+1 when it is possessed by n, is possessed by all finite numbers. By means of this principle, we prove that, if n be any finite number, the number of numbers from 0 to 12, both inclusive, is n+1. Consequently, if n exists, so does n+1. Hence, since 0 exists, it follows by mathematical induction that all finite numbers exist.

We prove also that, if m and n be two finite numbers other than 0, m+n is not identical with either m or n. It follows that, if n be any finite number, n is not the number of finite numbers, for the number of numbers from 0 to n is n+1, and n+1 is different from n. Thus no finite number is the number of finite numbers; and therefore, since the definition of cardinal numbers allows no doubt as to the existence of a number which is the number of finite numbers, it follows that this number is infinite. Hence, from the abstract principles of logic alone, the existence of infinite numbers is rigidly demonstrated.


2016-09-23
Challenge to the Mathematic and Logic Community
Reply to Andrew Wutke
So, simply, what you are saying without reiterating all of the statements above is, that since zero is something that is the absence of anything-- a concept that represents that.... then if you add something to that, it becomes, 1, 2, 3... and it can go on forever?  

It makes sense to me.  Also, if you take something from the additions that have accrued, you can become overdrawn, right?  So, "additive" infinity exists, as does that which "sets us back" (subtracts) to a place where eternity must be augmented by those constructive ones who potentially replace the digits that are taken by irresponsible succubus activity perpetrated by non-community beings, so that at some point the chosen additive scholars (holy theologians-- not a popular term in science or mathematics) might handle the deficit generators by absorbing the consequences of the actions of those that refuse to be responsible and move on in a harmonious "additive" eternity, those constructive ones having absorbed however unjustly (often at great personal cost) the deficits they did not create?  

Pardon me, but I'm a spiritual scientist, mathematician-- even very Catholic-- an odd and sometimes untenable combination.. ;-). Please take the time to consider the potential legitimacy of this post prior to rejecting it. Thank you for your consideration.  And God bless.