Need help understanding how certain mathemetical statements across the landscape can seemingly contradict (e.g. Cantor-Hume vs Euclid)

Question

Here are the main components to my understanding on this issue:

• Almost all of math can be given in a foundation of set theory
• Different math can seemingly contradict, e.g. in Euclidean geometry parallel lines never meet but they can in other geometries
• Those branches contradicting is okay because neither branch contradicts with the foundational theory. The branches at are "high levels", abstracted from the base a bit. These allow for the foundational theory to be agnostic to the higher level statement about parallel lines and its negation.

So far I can understand and swallow that two geometries can contradict.

But what if one of the statements is at/close to the foundational level? E.g. the Cantor-Hume principle that a "set is infinite if and only if it is equivalent to one of its proper subsets". Now I don't know exactly how close to the base axioms of ZFC the CHP is, but it is about sets, and we've been told the rest of math can be put in terms of sets/set theory. Then, further removed are mathematical concepts like measure. Measure, I've been told, makes use of Euclid's principle--the whole is greater than the part. Measure theory can be founded in ZFC set theory. But Euclid's principle, unlike before, seemingly can't be agnostic with the CHP. It's at a higher level, but contradicts with a statement at a lower level (seemingly). They conflict because an infinte set (a whole) is not greater than its proper subset (a part). This conflict I can't explain as in the first case.

So what's going on? Does Euclid's Principle contract the CHP like Euclidean geometry contradicts other geometry? Perhaps they don't contradict because the language of the two is sufficiently vague? And if they do conflict, how can we countenance that contradiction in the CHP case where the conflict seems to cut vertically across math instead of horizontally?

I'm asking here because I feel like I'm not yet capable of asking this in purely mathematical terms, and that there is potentially a philosophical side regardless.

Answer 1

Cantor-Hume principle is worded in the terms of cardinality: "set is infinite if and only if it has equal cardinality to one of its proper subsets". If you see it phrased in a different way, it is not precise.

What this phrase means in lay words is that a set is infinite if one can make a one-to-one correspondence between the set and one of its proper subets.

The Euclid's principle is phrased using numerosity, which is a generalization of the concept of measure. It says that the numerosity of a proper subset is less than the numerosity of the whole set. It defines infinite set as one which has numerosity greater than any natural number.

The both definitions of infinite sets, via numerosity or cardinality, coincide. A set, infinite in one sense is also infinite in the other sense.

Provided certain equivalence rules, numerosities of subsets of have total order and can be represented as elements of , a subset of surreal numbers. Numerosities of countable sets can be represented as elements of , which is the set of so-called, countable surreals.

Numerosities of some unbounded sets can change under shift, for instance, the numerosity of non-negative integers is greater than numerosity of positive integers by 1.They also depend on the metric or order of the space, so they can be defined differently on subsets of other spaces than .

Numerosity is a more precise notion of the set's size, as two sets with the same cardinality can have different numerosities, but two sets with the same numerosities inevitably have the same cardinality. If cardinality of one set is greater than the cardinality of another set, then its numerosity is also greater.