What meaning should we ascribe to infinite sets?

Question

When talking about things like integers, we can clearly ascribe a meaning to them. For example, 4 corresponds to "||||". The same is true of other finite objects.

This means, for example, that even if a statement isn't decidable in our given axiom system, its still clear that it is either true or false in an absolute sense. Take for instance the consistency of Peano Arithmetic. If Peano Arithmetic is inconsistent, we can write down a proof of 0=1. If not, then not.

This also means that axiom systems are either true or false. We can study PA plus the axiom that PA is inconsistent. This is a consistient system of reasoning, by Godel's second incompleteness theorem. Yet, it is incorrect. It asserts that there exists an object x which is a proof of 0=1 in PA, yet, when we put any particular object for x, the axiom is false.

My question is, what meaning should we ascribe to theories of infinite objects, such as set theory? For example, the axiom of choice is independent of ZF set theory, yet it seemingly very different the statement from the PA is consistient. Its not clear what it would mean for the axiom of choice to be true or false though. In fact, its not even clear what it would mean for the axiom of infinity to be true or false.

One way to approach the example of set theory would be to use a cumulative hierarchy, but that it itself dependent on ordinal numbers, an infinitary concept. If you had a theory of ordinal numbers, you could just use that theorem in conjunction with the cumulative hierarchy as your set theory.

EDIT: I guess I should say, is it possible to ascribe a meaning to infinite sets? If so, how would you do it. Can you give infinite concepts any meaning, or do they only have meaning relative to a formal system (which reduces them back to finitary objects, since the formal systems are finitary).

Answer 1

My question is, what meaning should we ascribe to theories of infinite objects, such as set theory? For example, the axiom of choice is independent of ZF set theory, yet it seemingly very different the statement from the PA is consistent. Its not clear what it would mean for the axiom of choice to be true or false though. In fact, its not even clear what it would mean for the axiom of infinity to be true or false.

There's really no way to answer your question unless you can elaborate on what you mean by ascribing meaning to sets. Sets just aren't the sorts of objects that have meanings, as standardly conceived. Linguistic objects, like words and sentences, have meanings. For example, taking a broadly Fregean view the word "Antartica" has a meaning that consists of a sense and a reference. The reference is the continent of Antartica and the sense is the mode of presentation of the reference (for example, "Antartica" and "The least inhabited continent" have the same reference but different senses). Now what meaning does {1,2,3} have? It certainly doesn't refer to anything; it's just a collection of objects. It doesn't seem to have a sense either. What, then, do you mean by a set having a meaning?

Secondly, do you really think it's unclear what it would mean for the axiom of choice to be true or false? What about the statement of that axiom confuses you? What do you think mathematicians are doing when they routinely use the axiom of choice? What are they doing when (less routinely) they prove theorems containing sentences like "Suppose the axiom of choice does not hold"?

Answer 2

Words have "meaning".

Set and number are words denoting abstract objects; thus, they are subject to allo the ontological and epistemological issue concerning abstract objects.

They are one of the modern aspects of the very old problem of universals, going back to Plato.

So, referring to your question : how is possible for a "universal" term (or concept) to have meaning ? We have to assume that - in order to have meaning - it has to denote some sort of "real" entity ?

This sort of questions is not related only to mathematical concepts: what means/denotes "magnetic field" ? and "social class" ?

Thus, imo, your question is more about "existence" than meaning :

what kind of existence we ascribe to infinite objects, such as (infinite) sets ?

Answer 3

One set of people who have actually worried about this are the Intuitionists, who strive for a real, answerable, standard for meaningfulness, in their mathematical objects, if one that has to be continually renegotiated. They believe that mathematical objects are only reliable to the extent they can be mapped onto some aspect of human psychology that can be captured clearly and communicated.

The general approach in Intuitionism is to consider all infinities mere potentialities and to avoid reifying them. The integers represent the potential of a process that simply does not stop. The continuum represents divisibility that simply does not stop. And so forth.

Considerations like the Axiom of Choice are replaced by an absolute prohibition upon completing any infinity. On weaker grounds, the standard goes back to Aristotle. But, since Kant's antinomy of space, we are more certain that reifying endlessness conjures up an internally inconsistent set of intuitions that we simply cannot handle adequately. Kant is free to say that means space is not real, but mathematicians would find that inconvenient. Still, his point is made, we need to avoid imagining that we can imagine too much, in such domains -- it gets us in trouble.

Instead of completed wholes, we must look at ininities as processes of evolution, with sets of rules, preferably identifiable states and transitions. So, for instance, in studying the ordinals, we are not addressing a trans-infinite class of sets, we are analyzing the variety of possible kinds of ordering -- there is no omega-to-the-omega-plus-seven, there are only rules that say how to compare things that are free-flowing sequences of integers, with noted exceptions that dominate all of those sequences. When we study cardinality, we cannot reify the orders of infinity, but we can explore the potential conflicts that humans should be aware of in the process of making a bijection.

This squares well with the more modern notion of Category Theory. What is important, in addressing any mathematical 'thing', is the potential interactions it might have with the other things of its kind. If those interactions are pinned down by stating rules about identified elements within it, that does not mean the elements or even the collection is 'real', only the composite behavior is. It just means it is convenient to imagine the object in either global or more local terms for axiomatic testing purposes.

Answer 4

David Hilbert in 1904 [...] wrote that sets are thought-objects which can be imagined prior to their elements. At request of the referee who asked what is a thought-object let me add: I understand it to be a thought about an object which may exist or not. Thus it is an electrochemical event in the brain or/and its record in the memory. In particular it is a physical thing in space time. Of course it is difficult to characterise any physical phenomena. But we have the ability to recognize thoughts as identical or different, just as we have the ability to recognize a silent lightning from a thunderous one. Hence I understand Hilbert's words as follows: mathematicians imagine sets which do not exist, but their thoughts about sets do exist and they can arise prior to the thoughts of most elements in those sets. [J. Mycielski: "Russell's paradox and Hilbert's (much forgotten) view of set theory" in G. Link (ed.): "One hundred years of Russell's paradox: mathematics, logic, philosophy", de Gruyter, Berlin (2004) p. 534]

I think that this quote says it all. We know that in the accessible part of the universe it is impossible to name infinitely many elements, for instance the natural numbers. Every physical realism will prove finitism. There is no largest number known, but there is a largest complexity of (mathematical) expressions, given by the number of atoms accessible to us.

Infinite sets do exist as names and illusions like the spaghetti monster or unicorns.

Answer 5

I might be wrong, but here is my stance:

This also means that axiom systems are either true or false. We can study PA plus the axiom that PA is inconsistent. This is a consistient system of reasoning, by Godel's second incompleteness theorem. Yet, it is incorrect. It asserts that there exists an object x which is a proof of 0=1 in PA, yet, when we put any particular object for x, the axiom is false.

You are mixing higher order logic concept (the axiom that PA is inconsistent) and first order logic concepts, in your sentence, which makes it hard to understand. You can put a particular object for x for which the axiom is true as the axiom that PA is inconsistent is by itself a proof that such x exists and therefore allows you to instantiate it, but in the higher order logic you are using, not in original PA. The fact that you try to put a particular x, for which you inevitably get 0≠1, is not relevant.

My question is, what meaning should we ascribe to theories of infinite objects, such as set theory? For example, the axiom of choice is independent of ZF set theory, yet it seemingly very different the statement from the PA is consistient. Its not clear what it would mean for the axiom of choice to be true or false though. In fact, its not even clear what it would mean for the axiom of infinity to be true or false.

I think I understand what you mean. You see a huge infinity gap between the infinite sets in ZF theory and the infinite set of properties of PA (including its consistency or inconsistency), and you would be right.

ZF theory (and PA) is meant to squash as much as possible of high order logic into first-order theories without becoming inconsistent. First order logic feel artificial and meaningless because it was designed to facilitate automated reasoning in the sense that by applying dumb rules over axioms we ends up with proofs we agree on. But we naturally use higher order logic as they are more succinct, more powerful, but also error prone to handle.

As such, ZF as a first order theory cannot describe the infinite set of properties of PA for example. You need to consider ZF as a higher order theory which kills the point of ZF. I would suggest the Simple Type Theory if you want an easy to use higher order theory.