[FOM] Improving Set Theory

[Arnold Neumaier]

On 09.01.20 19:10, Harvey Friedman wrote:
> ZFC has become the standard foundation for mathematics since about
> 1920. Alternatives have been proposed but not widely endorsed at least
> not yet.
> 
> I am particularly interested in what people think is lacking or is
> flawed about ZFC.

All meaning (and hence all understanding) is eliminated when
reducing mathematics to pure ZFC. Moreover, since the same
mathematical concepts can be (and are in practice) encoded
in ZFC in multiple (set theoretically inequivalent) ways,
math encoded in ZFC is different depending on who does it.

Are your real numbers Dedekind cuts of rationals or infinite
decimal fractions or equivalence classes of Cauchy sequences,
or....? It doesn't matter at all for the mathematics, and
shouldn't matter for the foundations.

Many (if not most) working mathematicians believe that ''the''
real numbers are unique, and not different objects depending
on how they are constructed. Constructions just give names
to numbers.

Thus ZFC is just a possible naming scheme for mathematics.
It captures nothing of its essence.

ZFC relates to mathematics like assembly language relates to
modern programming languages. Most mathematicians care as
little about ZFC as most programmers care about assembler
code. Like for a progrmmer, any of the many ways of
producing fast and correctly executing code is as good as
any other, so any of the many ways of reducing the purely
logical part of math to some logical system is as good as
any other.

Indeed, most proof assistants used to encode advanced
mathematics in fully formalized terms are not based on ZFC,
and are incompatible with each other.


Arnold Neumaier