Re: What is the 1st order formal system known as PA?



Rupert wrote: 
Nam Nguyen wrote:

Rupert wrote:

Nam Nguyen wrote:



What I propose is very much similar. There would be many
"PA[-ish]" theories under the proposal, all of which would share the
same unary [successor] function symbol "S".



So by a PA-ish theory you just mean an extension of PA?


No. Under my proposal, there is no lone PA theory to be
extended from.



There is a standard definition of PA and there's only one theory that
satisfies it.

Care to prove that there is only one theory *satisfying* the standard
definition of Peano axioms? Or is that because "Philosophically, it's
completely obvious", again?


Instead, there are many PA-ish theories all
having the same language L(PA), satisfying Peano axioms [which
means all have the same successor function symbol "S", among
other things.] Note that _here_, in "L(PA)", "PA" doesn't denote
a theory!



Well, obviously you're going to have to give a definition of "PA-ish"
theory if we're going to be able to know what you're talking about.

I've just given so, the very paragraph right above! Did you really read
that paragraph?

There's no such thing as an axiom we could not state. 

Would you care to provide a proof for your meta statement above?

The notion "we can state it" is not a mathematical one, so I can't give
a mathematical proof. Philosophically, it's completely obvious. An
axiom is just a certain kind of string of symbols. Any axiom of
reasonable length, we can state. I suppose an axiom might be too long
for us to state. But I don't think that's what you had in mind.

Regardless of why you don't think that's what I had in mind, would
you concede now that there are axioms we couldn't state [i.e.
anonymous axioms]?

If you don't, then your tendency of making assertions [or refuting
people's statements] without concrete proof, and of not following
what people have just said seems to make our conversation in this
thread hit a dead-end. And I'd probably have to leave it at that!



--
--------------------------------------------------------------------
The difference between landscape and landscape is small,
but there is a great difference between the beholders.
                                     Ralph Waldo Emerson
---------------------------------------------------------------------

.

Relevant Pages

  • Re: Set Theory: Should you believe?
    ... with a successor function, and with a successor relationship where ... one assumes in the axioms that the successor relationship is a function ... formulate the Peano Axioms which are non-equivalent. ... Bringing in these other treatments where "number" ...
    (sci.logic)
  • Re: Set Theory: Should you believe?
    ... with a successor function, and with a successor relationship where ... of formulating the Peano Axioms are essentially the same. ... logicians commonly refer to as Peano Arithmetic, is the "first order projection" ... infinitely many naturals and others are not. ...
    (sci.logic)
  • Re: infinity
    ... Not while discussing the consequences of the axioms as they actually are! ... > You never commented on my adjustment of the Peano axioms, ... IF you claim to have generated an infinite set with your ... I am applying only finitely many successor operations at ...
    (sci.math)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... axiom replaces Peano's 5 axioms.) ... I prove theorems of Godel, Rosser, Turing, Smullyan and you say I ... that you trumpet it to be, not simply its application to recursion ... Every number has a successor number: For every N printed in line 2, ...
    (sci.logic)
  • Re: infinity
    ... >>> The Peano axioms prohibit any complete Peano set from having a largest. ... You never commented on my adjustment of the Peano axioms, ... Well, after all, you ARE applying an INFINITE number of successor operations, ... IF you claim to have generated an infinite set with your stepwise difninition, ...
    (sci.math)