Peano arithmetic is enough, because Peano arithmetic encodes computation

https://math.stackexchange.com/a/5075056/6708

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This Topos Institute seminar (Kevin Carlson presenting) is an interesting topic -- do we actually need infinite sets to do mathematics? But, I think he takes a long and questionable route to the meat of the topic.

https://youtu.be/bHKvT1ZACLY

The fallacy here is that the answer to "why is there so much consensus in modern mathematics" cannot be a mathematical answer! It has to be grounded in something else: sociology, history, or psychology. It's all very well to point at the structural approach as a unifying point of agreement, but that by itself does not answer "why"?

It could be: humans have some set-sense like they have a language-sense, and so building things on sense connects to a lot of people. That might be JP Mayberry's point, in his appeal to "Euclidean set theory." But that's not a mathematical claim!

It could be: people who do not get on board with the structuralist approach don't succeed at being modern mathematicians.

It could be: we are living in an era that encourages that consensus instead of discouraging it, for reasons that will be evident only in retrospect.

I think part of the answer is that modern mathematics has achieved consensus by successfully eliminating "truth" as a topic of debate and instead making it a topic of study, in a very postmodernist way. You can insist to your dying day that you're a intuitionist or an ultrafinitist or whatever, and the only response you can get is mathematicians studying what is or isn't provable in your version of logic! It is no longer possible to disagree what mathematics is, because the modern conception swallows any such disagreement into a mathematical object.

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