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Topic: Just what is equality in mathematics, anyway?
Replies: 45   Last Post: Apr 20, 2012 5:56 PM

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 Joe Niederberger Posts: 1,659 Registered: 10/12/08
Re: Just what is equality in mathematics, anyway?
Posted: Apr 17, 2012 10:20 AM

>Either way, since he did use the replacement property of logic with respect to either equivalent or equal well-formed formulas

There are NO WFFs on display here Paul.

What formal system are you imagining you or Rudin is working in? That's the only context in which WFFs make sense.

And in such a system, one doesn't merely note that "arithmetical properties are the same" and then conclude "therefore we can identify"-- you have to have a formal derivation to get to those sorts of statements (which of course, would not be in natural language.)

>Note that he uses his prior statement that we can identify (a,0) with a to replace real a and b with complexes (a,0) and (b,0) in the first equality in the proof, using the replacement property of logic with respect to either equivalent or equal well-formed formulas.

OK, this is just silly and wrong. When Rudin says "equivalent" he means isomorphism. He is not saying he has logically equivalent WFFs up his sleeve. His treatment is informal. Also, as far as I know, he is not talking about the set theoretical constructions that I specifically stated I was talking about, in which an ordered pair (a,b) is shorthand for { a, {a,b} }.

But you think you do, so show em. Show your cards.
Where's the WFF?

Joe N.

Joe N.