Existence and Numbers as Sets of Properties

Existence and Numbers as Sets of Properties

In 1^st-order number theory one proves, for example, “5 exists”, expressed

(1) (there exists x)(x = 5).

But what happens if

(2) an existence statement about a natural number is a strictly 2^nd-order property

(3) a natural number is defined as a set of properties (that the number would have had, so to speak)

Both are inconsistent with (1), or at least arguably so, and both have a surprising amount of philosophical motivation. To get an idea of what (2) might mean, “5 exists” might be replaced with “the properties of 5 are instantiated” (though this is only one interpretation of (2)). (3) is known as a bundle-theory, as opposed to a substance-theory. In a bundle-theory the number 5 is really just a bundle of properties, and there is no other entity that 5 refers to.

The question is, is there anything to be gained by implementing (2) or/and (3) in a mathematical theory?

I don't have the sophistication to answer this question on the mathematical side, or the philosophical side, so I'll speculate.

Part 1 (1) versus (2)

Let N1 be 1^st-order number theory with parameters {for all, etc.} as in (Enderton).

I'll stay with the relatively simple intuition above: “the number 5 exists” is to be replaced with “the properties of the number 5 are instantiated”. How do you define the set of properties of the number 5?

The set of properties of 5 could be defined as the (“provable”?) collection of closed formulas in N1 that contain at least one instance of 5. Call this “characteristic” set C_N1(5). This set is probably too big, at least in the sense that N1 contains the 1^st-order existence assertions we're trying to avoid. So, informally, define

N2 is just like N1 except it doesn't contain existence-statements (or for-all statements)

N3 is just like N1 except it doesn't contain existence-statements, and it does not have strong negation, but it does have for-all statements (?)

Anyway, there are many theories smaller than N1 one could start with.

Consider N2's 2^nd-order language. Does it have the existence-quantifier? I don't know. Does one need a new “instantiation” quantifier I, such that if p₁ is a property of 5,

Ip₁ iff one would normally prove 5 exists

But I might not be necessary, because 2^nd-order instantiation looks (to me) like it behaves like 2^nd-order existence.

The point is this. If you'd normally say something exists, you can now say there is a (maximal?) collection of properties (its characteristic set) that must have certain conditions on it. To begin with all of the properties of something that we would normally say “exists” must be consistent with each other, e. g. if p and q are in a characteristic set, then the property p and q must be in it. Call this set of conditions CondEx. Now potential “things” can be recognized by this condition on the corresponding set.

(Decidability issues.)

Consider the collection of all sets of 1^st-order properties. Which sets have CondEx? Are these exactly the ones that correspond to what we'd normally call (extant) numbers? Do the non-standards play a new role?

What is the definition of CondEx? Certainly there are more than just consistency criteria that characterize existential things. The properties a unicorn would have, if the unicorn existed, are consistent with each other.

The next question is how can the conditions CondEx be relaxed (or strengthened)? Suppose there are other sets that meet a condition weaker than CondEx, do they exist, but to a lesser degree? (This from the philosophical literature.)

One issue is that if a number exists in the sense of (2) you'd expect distributivity to hold for all of its properties. But it does not always hold in a quantum context. In one variant of quantum logic, it may be true that an electron

“has a y-spin up and has an x-spin up or down”

but it is not then true that the electron

“has a y-spin up and an x-spin up, or has a y-spin up and an x-spin down”

yet the electron exists. (I'm not saying this is a paradox, just that it's interesting.)

In fact, suppose that, in some generic sense of ontology, “a exists” and “b exits”. It is a further assumption on the nature of ontological existence that there are “two things” in any sense, or that a and b can be compared in some way. Physicists currently consider disjoint universes that have different, though related, physical laws. What are the philosophically motivated alternative definitions to CondEx? i.e. other conditions on sets of properties of something that might reasonably exist.

Let C_N2(5) be the properties of 5 starting with N2, properties being provable sentences (and “provably” so, if necessary) containing at least one 5. One could argue that a more complete characterization of 5 can be given by it's 3rd-order properties, call it C(C_L2(5) ). Either the 2^nd-order language has the existential quantifier, or it doesn't, whatever. This process can be continued indefinitely. The question is, is there a point at which the collection of n^th-order (or higher) properties of 5 starting with N2 is isomorphic to the collection starting with N1?

If so, then 1^st-order existence in arithmetic is in some strong sense dispensable, and it is immune to many philosophical objections.

Voevodsky at the IAS conjectured (http://video.ias.edu/voevodsky-80th ) that 1^st-order arithmetic is inconsistent. He explained how and why this could happen. He also points out that it would nevertheless remain true that if you have 2 physical apples, and you have 3 more physical apples, you will then have 5 physical apples. So there is some “physically justified” proper subset of arithmetic that is “necessarily” consistent, even if arithmetic itself is inconsistent. The question is: what is the extent of this physically justified subset?

A physically justified subset will have objects and operations. (I tried getting around that, thinking objecthood isn't so straightforward in quantum field theories, but the guys at MathOverFlow pointed out the flux integrals around disjoint sets sum additively. So far as I can see, that shows the quantum case is like the classical case, but doesn't settle what the classical case is.) The objects are supposed to be the things that the theory asserts exist. Given the notions of existence above it might be possible to delineate the maximal physically justified subset of arithmetic.

Part 2 (1) versus (3)

According to (3), 5 is to be defined by its properties. These properties are initially sentences in which 5 occurs. But some of these will be like 2 + 3 = 5. To be consistent, one wants 2 and 3 to be defined by sets of properties too. But some of the properties of 2 and 3 will include 5. Thus 5 will really be a set of properties of properties of properties of... This is non-wellfounded behavior (Aczel, Barwise and Moss). So far as I can tell, most “circular” phenomena like this can be modeled by non-wellfounded sets, though there are some would-be definitions that are inconsistent.

For example consider

(4) 2 + 3 = 5

and suppose one of the properties of 3 is

(5) not(2 + 5 = 3)

In view of (2), (4) and (5), and depending on the iteration process, informally speaking, stuff like this will come up in the beginning:

(6) {1st-order properties of 2} + {not(2 + 5 = 3), other 1st-order properties of 3} = 5

It looks to me like the non-wellfoundedness, at least, is not a problem.

Part 3

What happens when (2) and (3) are combined? Is it necessary to combine them for philosophical reasons?

What happens when they are combined in other mathematical contexts?

Set Theory. Is the definition of CondEx the same in set theory as in arithmetic? Do the universes of L, ZFC, whatever, get bigger?

Some version of Comprehension might be recoverable. The standard problem is one can define a set R that both contains and doesn't contain itself. But if R doesn't exist in the first place, it can't cause any problems for the universe.

Finally, if questions like these can be given definite answers they would impact the philosophical discussion.