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After Cantor, mathematicians realised that infinities can be graded by size (cardinalities). But just as we view the naive infinite as bigger than any finite number, is there an 'infinite' greater than any infinite?

But then the same pattern can carry on, in which case we are actually no nearer the absolute infinite than we were to begin with, that is Cantors mathematics of the infinite cannot do real justice to the idea of the Infinite.

To be more explicit, the Large Cardinal Axioms extend ZFC by postulating ever higher 'Cardinalities'. "However the observation that large cardinal axioms are linearly ordered by consistency strength is just that, an observation, not a theorem. (Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense)...Saharon Shelah has asked, "[i]s there some theorem explaining this, or is our vision just more uniform than we realize?" Woodin, however, deduces this from the Ω-conjecture, the main unsolved problem of his Ω-logic."

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"is there an 'infinite' greater than any infinite?" Mathematically, no. "Cantors mathematics of the infinite cannot do real justice to the idea of the infinite" Alternatively, one could hold that set theory changed our notion of infinity, in that it corrected a flawed folk understanding of it. –  DBK Dec 1 '12 at 12:22
    
Just in passing: I think Cantor was religious, and held that the absolutely infinite was God. –  Joseph Weissman Dec 1 '12 at 14:23
    
@DBK: I'm not so sure of that. It showed that the infinite have structure, but the absolute infinite, what you might call the folk infinite is still out of reach. Look at the proposed list of large cardinal axioms which extend the range that ZFC can reach. There is so far a finite list and they can be linearly ordered. –  Mozibur Ullah Dec 1 '12 at 15:54
    
@Weissman: I think you're right, one could also infer from that he also believed the absolute infinite was also out of reach from mathematics. One can always go higher. So the folk idea of the infinite is corrected to the idea of the absolute infinite. Is there any connection with Hegels notion of the absolute infinite here? –  Mozibur Ullah Dec 1 '12 at 15:56
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Actually, what Cantor specifically called the "Absolute Infinity" (and went mystical about, as @JosephWeissman noted) can be treated mathematically. See also my answer to another question here. –  DBK Dec 1 '12 at 17:37
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2 Answers

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There is no "absolute answer" as to whether there is an absolute infinity, because whether or not you can have an absolute infinity is a function of what mathematical formalism you use.

  1. If you use ZF set theory (with or without the Axiom of Choice), you can for any infinite cardinal construct one which is larger by diagonalization. So even though not all infinities would necessarily be comparable, there would always be one cardinality which was demonstrably larger.

  2. If you use NBG set theory — a conservative extension ZFC — then there is an absolute infinity: the cardinality of proper classes, which in the usual construction of these things would be identified with the proper class of all von Neumann ordinals. Of course, this cardinality wouldn't exist as an element in the cardinality function using any conventional set- or class-theoretic construction of functions, because this would require the infinite cardinality to be an element of a class, which is impossible for proper classes by definition. But one could certainly make a statement, in NBG theory, about whether or not a class has a bijective mapping to the universal class, which would certainly indicate that it had the largest cardinality.

One criticism that you might make is that you could always make a model for NBG in some other set theory in which there are no proper classes. (This is essentially what you point out with inaccessible cardinals). Because one can always extend beyond any one model in which there is a largest cardinality, one can say that in the subject of mathematics — not in any one model, but in the discipline as a whole — there is no absolute infinite. But of course, if you do not fix any theory whatsoever, nothing whatsoever can be absolute. Non-standard models of the Peano Axioms subvert our usual notions of the natural numbers, so that we could not even say with any confidence that there is an absolute notion of a natural number. There isn't even an absolute notion of what it means for a cardinality to be infinite, if you allow the changing of models as you like. You can remedy this by fixing some class of set theories you wish to discuss, but this is just fixing some particular theory — a theory of set-theories — in the same way that selecting NBG or ZFC would be fixing a theory; your answer will depend on what theory you fix.

In order to give any sort of answer, you must fix your domain of discourse sufficiently to give an answer. So depending on how you want to fix the terms of your question, the answer is either

  • "yes there is" (as in NBG set theory, or a meta-theory which is like it),
  • "no there isn't" (as in ZFC set theory, or a meta-theory which is like it), or
  • "we can't answer the question" (because the terms haven't been defined well enough).
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John D. Barrow's The Infinite Book contains the following passages:

Cantor's most dramatic discovery was that infinities are not only uncountable, they are insuperable. He discovered that a never-ending ascending hierarchy of infinities must exist. There is no biggest of all that can contain them all.

also

Mathematical 'existence' meant only logical self-consistency and this neither required nor needed physical existence to complete it.

and (this after a mental breakdown that Cantor suffered in 1884)

Indeed, Cantor said, it would have diminished God's power had God only created finite numbers.

The author John D. Barrow is a Cambridge mathematician and physicist who has also written a play called Infinities where Cantor makes an appearance.

I sense in your questions a certain discomfort with the mathematical concept(s) of infinities: For instance if so, you may also find the chapters on Leopold Kronecker's finitism and on the replication paradox (Universes Where Nothing is Original in The Book of Universes by the same author) rewarding.

My own current view is that it is helpful to recall from mathematics that infinite is the inverse of zero. I understand there were many cultures in human history where the concept of zero was unknown or uncertain (try writing down zero as a Roman numeral, for instance), just like there were cultures where words for certain colors were absent (the adjective blue in Mediterranean Ancient Greece, of all places).

Nowadays the possible paradox "How can nothing be something?" does not usually strike us as dangerous when we think of zero. So maybe some sort of cultural shift will bring a similarly new understanding of (the idea of) the infinite in a linguistic, philosophical, or even mathematical sense.

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I don't have any problems with Cantors Infinities, nor with Large Cardinals, going further than Cantor did. I also know about the Ordinals which is a finer-graining of Cantors Infinities. But my own naive concept of infinity: is that which cannot be gone beyond. It doesn't look like the mathematicians have come even close. –  Mozibur Ullah Dec 9 '12 at 4:53
    
@Mozibur Ullah sorry for my misassumption. I looked up your profile and now realize that you must be already quite well versed in this field. Could you please indicate a bit more about what you refer to as the absolute infinite (perhaps in terms of published work or articles on the web. Or is it just the naive concept?) This may also further improve the question. –  Drux Dec 9 '12 at 6:50
    
@MoziburUllah, your idea of infinite is indeed naive, but that's natural because the concept goes beyond our current intuition. It takes time to develop an intuition for a new concept. There will be a time when everyone will be comfortable with the idea of infinite. –  Cantor Dec 9 '12 at 12:25
    
@Drux: Please see my answer to this question:Cantor & Infinities. In short the naive idea of infinity, is a reflection of a philosophical concept 'the Absolute Infinite', and not actualizable in mathematics. No matter how far we go in formalising the infinite, there is always further to go, which is why Cantor himself became mystical about it. When I first read about Cantors taming of the infinite in my teens, I got all excited about it; when I realised he'd got nowhere near, I was disappointed and annoyed –  Mozibur Ullah Dec 9 '12 at 13:13
    
@Drux: and irritated at the hype surrounding his achievements (this is not to belittle them). Writers on the Infinite should be more careful... –  Mozibur Ullah Dec 9 '12 at 13:15
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