Would an axiom system for finitism, plus an infinite set…
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Would an axiom system for finitism, plus an infinite set of indexes, work in practice?
I did read this article https://www.quantamagazine.org/what-can-we-gain-by-losing-infinity-20260429/ on Quanta Magazine about ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism in mathematics, and I've got curious about it.
After reading some Wikipedia articles about finitism, and noticing that there is some research on it, I thought: what if one accepts infinities on mathematical reasoning, but only finite sets as mathematical objects? For instance:
Remove from ZFC the axioms of infinity and choice (finite choice is still possible), and add a single, special, countably infinite set of *indices*, with induction a la Peano, but no operations beyond the successor one. Any general results for operations on the (finite, bounded) sets of integers and rationals could be proved via a back-and-forth with the set of indices. Finite sequences would be a function from a subset of the set of indices to a finite set of natural numbers, and then to the domain of sequence values.
Such a scheme would allow one to recover some of the theorems lost by removing the notion of infinity, while allowing for only finite sets.
Is that a viable axiom system? Would it work in practice?
#science
I did read this article https://www.quantamagazine.org/what-can-we-gain-by-losing-infinity-20260429/ on Quanta Magazine about ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism in mathematics, and I've got curious about it.
After reading some Wikipedia articles about finitism, and noticing that there is some research on it, I thought: what if one accepts infinities on mathematical reasoning, but only finite sets as mathematical objects? For instance:
Remove from ZFC the axioms of infinity and choice (finite choice is still possible), and add a single, special, countably infinite set of *indices*, with induction a la Peano, but no operations beyond the successor one. Any general results for operations on the (finite, bounded) sets of integers and rationals could be proved via a back-and-forth with the set of indices. Finite sequences would be a function from a subset of the set of indices to a finite set of natural numbers, and then to the domain of sequence values.
Such a scheme would allow one to recover some of the theorems lost by removing the notion of infinity, while allowing for only finite sets.
Is that a viable axiom system? Would it work in practice?
#science
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