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How is that different from claiming, say, there's an integer betwen three and four, oh we can't construct it but that's not a problem, gonna use it in the proof
by rini17 4y ago
How is that different from claiming, say, there's an integer betwen three and four, oh we can't construct it but that's not a problem, gonna use it in the proofs anyway?
- jbnicolai 4y agoThat's essentially what axioms are. Wether it's reasonable or not depends on your intuition of both axiom & implications. This is where the axiom of choice is interesting; many people feel that it seems intuitively reasonable, but the implications are counterintuitive (e.g. banach-tarski)
- contravariant 4y agoWell an integer between three and four is going to cause some contradictions, depending on your definition of 'integer'. You can assume there exists a number between 3 and 4 though, no problems there.
- jacobmartin 4y agoThis is an interesting question that gets to the heart of the constructivism debate, which I won't touch on here. As for the integer example, the integers are all easily (finitely) constructed (successors to 0 and their additive opposites), so it isn't a consistent concept to say "This is an unconstructable integer." Sets, on the other hand, are not necessarily always constructable, unless you only take the constructivist axioms.
- xigoi 4y agoAn axiom has to be consistent with other axioms (that is, adding it doesn't introduce a contradiction).
- xen0 4y agoThat would let me derive your mother's phone number. https://xkcd.com/704/ https://xkcd.com/704/ It's different because there's nothing that says such a choice function _cannot_ exist (short of an axiom that says explicitly so).