
Re: An alternative Axiom of Infinity?
Posted:
Feb 4, 2014 1:59 PM


On Tuesday, February 4, 2014 1:22:28 PM UTC5, Peter Percival wrote: > Dan Christensen wrote: > > > On Tuesday, February 4, 2014 12:39:46 PM UTC5, Peter Percival wrote: > > > > > >>> Get with the times, Peter. > > >> > > >> > > >> > > >> What's sauce for the goose is sauce for the gander, so 'Cauchy to you'. > > >> > > >> But far more significant than the passage of time is that you seem not > > >> > > >> to understand the difference between a theory and a definition. > > > > > > Again, Cauchy's recommendation that 0^0 is be left undefined is still widely accepted today, whereas Zermelo's original definition of the successor function is not, not even among set theorists. Get over it, Peter. > > > > It has got nothing to do with what is accepted. ZF set theorists may > > well adopt the von Neumann definition of ordinal number, but that is a > > choice they make, the definition doesn't come automatically with the theory. >
Definitions are NOT divinely inspired; they are devised by people. If they are useful, they will continue to be used; otherwise they won't be. And Zermelo's original definition no longer seems to be useful to set theorists.
I don't know why you keep going on about this, Peter. Even under the original definition, you had "junk theorems" in ZF like 2 being an element of 3. In this regard, you can't rescue ZF with this misplaced nostalgia for long disguarded definitions.
> > > It is real and complex 0^0 that is undefined, not natural 0^0. >
That issue has been hotly debated and finally settled here: 0^0 is undefined on the natural numbers as well. Deny it if you like, but readers can see for themselves in the thread, "Constructing the Addition, Multiplication and Exponentiation Functions from Peano's Axioms," among several others here.
Dan Download my DC Proof 2.0 software at http://www.dcproof.com Visit my new math blog at http://www.dcproof.wordpress.com

