Start with this intuition:
- Every sentence of first order logic with the successor predicate s(x,y) (which says that x is the natural number succeeding y) is determinately true or determinately false.
We learn from Goedel that:
- No finitely specifiable (in the recursive sense) set of axioms is sufficient to characterize the natural numbers in a way sufficient to determine all of the above sentences.
This creates a serious problem. Given (2), how are our minds able to have a concept of natural number that is sufficiently determinate to make (1) true. It can’t be by us having some kind of a “definition” of natural numbers in terms of a finitely characterizable set of axioms.
Here is one interesting solution:
- Our minds actually contain infinitely many axioms of natural numbers.
This solution is very difficult to reconcile with naturalism. If nature is analog, there will be a way of encoding infinitely many axioms in terms of the fine detail of our brain states (e.g., further and further decimal places of the distance between two neurons), but it is very implausible that anything mental depends on arbitrarily fine detail.
What could a non-naturalist say? Here is an Aristotelian option. There are infinitely many “axiomatic propositions” about the natural numbers such that it is partly constitutive of the human mind’s flourishing to affirm them.
While this option technically works, it is still weird: there will be norms concerning statements that are arbitrarily long, far beyond human lifetime.
I know of three other options:
Platonism with the natural numbers being somehow special in a way that other sets of objects satisfying the Peano axioms are not.
Magical theories of reference.
The causal finitist characterization of natural numbers in my Infinity book.
Of course, one might also deny (1). But then I will retreat from (1) to:
- Every sentence of first order logic with the successor predicate s(x,y) and at most one unbounded quantifier is determinately true or determinately false.
I think (7) is hard to deny. If (7) is not true, there will be cases where there is no fact of the matter where a sentence of logic follows from some bunch of axioms. (Cf. this post.) And Goedelian considerations are sufficient to show that one cannot recursively characterize the sentences with one unbounded quantifier.
4 comments:
Can't we have a concept of the natural numbers with only a finite number of axioms using second-order logic?
Technically, yes. However, I am one of those who think this just moves the bump under the carpet to a new place--the question of how we interpret the second order quantifiers, which is essentially the same question as that of what the class of sets is. (We have a choice between Henkin semantics and standard semantics for second-order arithmetic. If we choose Henkin semantics, we have indeterminacy because we can vary what the subsets are that the second-order quantifiers vary over. If we choose standard semantics, we have just shifted the bump under rug to one of how we gain reference to "the sets".)
I'm not really sure if I understand the difficulty in gaining reference to "the sets" - they are just collections of the first-order objects of discourse. (I do know about Henkin vs standard semantics, but the former has always seemed to me to be just a way of modelling 2nd-order logic in 1st-order language, for the purpose of being able to use 1st-order logic's formal properties, rather than a serious candidate for 2nd-order semantics.)
If the first-order objects of discourse include the naturals, then we have the question of what are collections of naturals. There are uncontroversial such collections, namely ones of the form { x : x is a natural number and F(x) } for a formula F. But it is implausible to think that these collections exhaust the collections of naturals. (For the sake of intuition, think here of the "math tea" argument--which has some technical difficulties, so I'm only using it for intuition--which notes that there are only countably many such formulas but uncountably many collections of naturals.) But how we gain reference to the concept of a "collection of naturals" that goes beyond the uncontroversial ones is hard for me to see.
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