Are there foundational theories of mathematics without constants?

Question

One of the simplest useful axiomatic system in mathematics is the Peano Axioms for the natural numbers. Notably it has only a single constant (the number zero).

One might suppose that one can do away with this single constant by considering an axiomatic framework for the integers which does not have a distinguished starting value. The usual construction for the integers, which goes via the natural numbers seem to deny that possibility. (Alternatively one might show that it has a unique model, is this possible)?

ZFC also has constants via the axiom of infinity - that there is an infinite set - an alternative way of seeing this, is that in topos theory, which is a generalised set theory, the replacement for the axiom of infinity is the natural numbers object which is the categorical equivalent of the peano axioms, so we're back to the first case mentioned above.

Now, a topos does not neccessarily have an natural numbers object, so perhaps that could count as such a theory, when considered foundationally; or since toposes are described via category theory, perhaps category theory when considered foundationally is such a theory. Is this in fact true, or is in fact a constant smuggled in via some other (sneaky) way?

Of course, there are theories without constants - just take any theory and drop any references to constants; we can even find useful examples - a heap which essentially models a group with its conjugation action (and is also, amusingly enough, a natural example of a mathematical structure with a ternary rather than binary operation); the question I'm after, though, is that it must be of foundational character - like ZFC, Category Theory or the Peano Axioms.

Answer 1

I think that none of the standard foundational theories in mathematics make an essential use of constants.

In the example of ZFC, there are no constant symbols in the language, and the official formal language of set theory is commonly taken to have only the set membership relation ∈ and equality, with no constants. As Niel says in the comments, the axiom of infinity asserts the existence of an object (an inductive set), but this is an existence assertion that can be made without necessarily introducing any constants.

In the case of arithmetic, although the usual axiomatizations of PA involve two constants, one for 0 and one for 1, in fact both of these can be omitted without sacrificing the expressibility of the theory. The reason is that 0 is a definable object, for it is the unique natural number that is not S(x) for any x, and therefore we do not need the constant symbol 0 to refer to it. Similarly, the number 1 is definable as the successor of the unique object that is not a successor, and so we do not need 1 in in the language.

So it seems that the answer to your question is that yes, indeed, there are foundational theories of mathematics without constants, and the standard foundational theories of arithmetic and of set theory can be undertaken without constants.