Timeline for Can infinity shorten proofs a lot?
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| Apr 19, 2012 at 17:31 | comment | added | Neil Toronto | Technically, ZFC doesn't have induction until the axiom of infinity is added. Point against this line of argument: most mathematicians don't really work in ZFC, but in a pidgin higher-order logic with sets, in which induction is probably axiomatic. Point for this line of argument: it probably matches the audience better, who (I would guess) don't distinguish between the potential infinity of induction and the completed infinity of the naturals. | |
| Feb 2, 2012 at 3:48 | comment | added | Halfdan Faber | Induction does not reach actual infinty, but could be viewed as an example of reasoning with potential infinity. Particularly note that from $IsFinite(0)=True$ and $IsFinite(x) −> isFinite(x+1)$ we prove by induction that all natural numbers are finite. | |
| Nov 18, 2009 at 10:01 | comment | added | gowers | I'm not sure I count induction either. It seems to me that one is not relying on infinity in a serious way to sum the first million cubes: one is saying, "If it's valid for 1 then it's valid for 2, and if it's valid for 2 then it's valid for 3, and ... and if it's valid for 999999 then it's valid for 1000000." That is a ridiculously long proof, but induction allows one to shorten it by giving a rule for when it's OK to put in the "..." (A more formal way of making the point is that the axiom of infinity isn't part of first-order Peano arithmetic.) | |
| Nov 18, 2009 at 8:49 | history | answered | David Corfield | CC BY-SA 2.5 |