I suspect there must be some concepts that math takes for granted (there has to be a starting point).
For example, after spending some time thinking about it yesterday, I wondered whether most of math could be produced from the concepts of
- Negation
- Identity
- Cardinality
- Ordinality
- Sethood
- Concepts
- Universality
It seemed to me that other basic concepts might be derivable from those concepts. For an example of what I mean, below I wrote out how I thought some of those derivations might proceed, roughly.
A. The universal set - from 6,5,7 - the set of all concepts
B. Complement - from 1,2,4,5,A - the set of concepts in the universal set and in a subset of it that are not identical.
C. Difference - from 1,2,4,5 - the set of concepts in two sets that are not identical.
D. Natural numbers - from 3,5 - the cardinal of sets (e.g. |{ {},{{}} }| )
E. Less than - from 1,2,5,D a subset of a number that is not identical to it.
G. Intersection - C,2,4,5 - from the set of the concepts that are identical and not of the ones that aren't.
H. Union - 4,5 - the set of the concepts of two sets.
I. Addition - 5,3,H,D - the cardinal of a union of two sets.
The question:
What are the fundamental concepts that we must (or, presently) take for granted when we do math?
Thank you.
Well, I think the correct answer to your question is even more parsimonious than the one you' ve given. The whole of math is reducible to standard set theory such as ZFC. ZFC can be expressed in a first-order language whose only non-logical constants are $\in$ and $=$ (in second order logic we can even dispense with identity as a primitive and define it via quantification over sets). So the only primitives needed for doing math are membership and identity (between sets).