On Sunday, 2 March 2014 16:32:27 UTC+2, Dan Christensen wrote: > > On Sunday, 2 March 2014 07:06:44 UTC+2, Dan Christensen wrote:
> 9 hours ago, your boasted, "I gave you 100% rigorous axioms that produce ALL rational numbers.... So, in five steps I have derived the concept of number for you."
Indeed I did. :-)
> The above statement was the first of these five "steps." Now you are saying this statement wasn't derived after all, but was an "axiom" or a "definition."
It's an axiom, and that's what I claimed. Nothing has changed and nothing will change.
Now, you have yet to respond to my retort:
Which axiom did you use to arrive at Peano's first axiom? I am asking you this to expose your folly, but you can't seem to get it.
So, if "0 is a natural number" is a "rigorous" axiom, how did Peano arrive at it? Which axiom did Peano use? And indeed, what is a "natural number"? After all, one can't just say things.
In fact, there are thousands of mathematicians (myself included) who do NOT consider 0 to be a natural number, but a whole or counting number. Nonetheless, I will desist from being a b1tch and say that it does not really matter with respect to the argument. What DOES matter is that Piddly Peano assumes the natural numbers exist, and what's more, that 0 is one of them! So, again I ask you:
Which axiom did Peano use to arrive at "0 is a natural number"?
If you can answer this, I will figure out a response for you. Otherwise I suggest you stop being a moron like you know who on this forum (W. of Oz).
> So, we are back to square one. We are STILL waiting for you to derive even one result from your "axioms." Again, I suggest you try to derive the most elementary result possible from your "axioms" ALONE: that there exists a number other than 0. Forget about "common sense" or "what every 2-year-old knows." You can use only YOUR OWN "axioms", John Gabriel.
I derived the entire RATIONAL NUMBERS. What have you shown me with Peano's rot? That it fails in the very first "axiom". You cannot assume the properties of those objects you plan to "define" through your axioms.
Ok, let me educate you a little. You cannot just define anything you please. A definition must be well formed. It must be REIFIABLE. Look up the word reify because chances are you DON'T know what it means. Most of you do not.
Reification, whether tangible or intangible helps us to quality control any definition. Axioms are not definitions but they still have to be well formed, for otherwise no one can understand them, and hence they are the antithesis of themselves! :-)
Moreover, you may NOT assume the prior existence of objects whose properties you did not prove.