# What concepts does math take for granted?

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

1. Negation
2. Identity
3. Cardinality
4. Ordinality
5. Sethood
6. Concepts
7. 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.

• Rules of logic; induction; "set". – Asaf Karagila Feb 1 '14 at 14:02
• For instance "Has the same cardinality" is a relation between sets expressed in terms of the existence of a function with certain properties, a function which is itself a set. – Kevin Arlin Feb 1 '14 at 14:05
• Hal, recall that all that which is definable in $\sf ZFC$ is definable from the rules of logic and $\in$ (which itself defines "set" along with the axioms of set theory). That includes the term "cardinal". – Asaf Karagila Feb 1 '14 at 14:17
• That is the greatest appeal of set theory, for me, that from just one binary relation and a few axioms, one can do pretty much everything. – Asaf Karagila Feb 1 '14 at 14:23
• Hal, one can write the formula $F(x)$ stating that $x$ is a set of ordered pairs (choosing a way to encode ordered pairs into sets, e.g. Kuratowski's definition $(a,b)=\{\{a\},\{a,b\}\}$ or otherwise) satisfying the condition of a function. Then one write $D(f,x)$ saying that $f$ is a function and $x$ is the domain of $f$, similarly $R(f,y)$ for $y$ is the range of $f$, and $Inj(f)$ for stating that $f$ is an injective function. Then one says that $|A|=|B|$ if $\exists f(D(f,x)\land R(f,y)\land Inj(f))$. – Asaf Karagila Feb 1 '14 at 14:26

## 1 Answer

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).

• (1) In the modern approach, $=$ is part of the underlying logic; (2) we can dispense of $=$ without leaving first-order logic. Define $a=b\iff\forall x(x\in a\leftrightarrow x\in b)$. Congratulations, you have now a working internal first-order definition of $=$ (assuming the axiom of extensionality). – Asaf Karagila Feb 1 '14 at 15:00
• Ad (1): Part of the modern approach is something called 'first-order logic without equality', where the symbol = is treated as a non-logical constant and interpreted as some equivalence relation. Models, where = is interpreted as the identity relation can then be singled out to be the normal ones. This approach obviously generalizes the more orthodox one you have in mind. – Jon Feb 1 '14 at 17:01
• To my best of knowledge, the addition of $=$ into the logical symbols is relatively recent. – Asaf Karagila Feb 1 '14 at 17:05
• (For example, if you look at old papers on set theory, they all explicitly define equality via $\in$. Only in newer texts equality appears as part of the logical symbols.) – Asaf Karagila Feb 1 '14 at 17:11
• Ad(2) Your definition reduces identity to the symbol $\in$, that neither is part of every first-order signature nor is it always interpreted as membership. So, you haven't provided a general first-order definition. – Jon Feb 1 '14 at 17:14