```Date: Jan 30, 2013 10:47 AM
Author: Dave L. Renfro
Subject: Re: Proving a definition of multiplication (wrong) by induction

Jonathan Crabtree wrote (in part):http://mathforum.org/kb/message.jspa?messageID=8185797> a = a added to a 1 time (the proposition)> > a does NOT equal a + aI don't see anything of a mathematical concern here. You'resimply arguing over the English-language meaning of the phrase"a added to a b times". Here are some examples of the statementsyou're dealing with:P(1) represents "a = 1*a"P(2) represents "a + a = 2*a"P(3) represents "a + a + a = 3*a"P(4) represents "a + a + a + a = 4*a"Each of these can be proved without mathematical induction(in an appropriate formal setting).We can also prove (in an appropriate formal setting) things like:"P(1) and P(2)""P(1) and P(2) and P(3)""P(1) and P(2) and P(3) and P(4)"This is simply because, in propositional logic, we have things like:A,B |- (A and B)A,B,C |- (A and B and C)A,B,C,D |- (A and B and C and D)That is, assuming A, B you can obtain "A and B". I say "obtain" because,depending on the logical system you're working in, this might be anaxiom. Of course, a 1-line proof consisting of an axiom is technicallya proof of that axiom (in the sense that "proof" is understood in formallogic), but I think there is less misunderstanding in the present contextif I say "obtain" rather than "prove".The previous statements can be rewritten as follows:(for all n in {1, 2})(P(n))(for all n in {1, 2, 3})(P(n))(for all n in {1, 2, 3, 4})(P(n))Each of the statements just above can be proved without mathematicalinduction.To give a more complicated example, we don't need mathematicalinduction to prove the following (in an appropriate formal setting,ultrafinitistic views excepted):(for all n in {1, 2, 3, ..., 10^10000})(P(n))However, you do need mathematical induction to prove [*]:[*] (for all n in {1, 2, 3, ...})(P(n))Incidentally, being able to prove [*] (in an appropriate formalsetting) is one thing, while proving that one can't prove [*]without mathematical induction is another thing. The "anotherthing" requires a bit of formal logical training just to correctlyunderstand what it means to prove that something can't be provedwithout mathematical induction.Now part of the problem in proving [*] by mathematical inductionis to assign an unambiguous meaning to the statement P(n) for eachpositive integer n (what you seem to be hung up on), but this is ameta-language issue and not something that lies within the formalsystem that the proofs take place in.Dave L. RenfroMessage was edited by: Dave L. Renfro
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