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Topic: A finite set of all naturals
Replies: 32   Last Post: Aug 15, 2013 4:07 PM

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namducnguyen

Posts: 2,699
Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 13, 2013 8:33 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 13/08/2013 10:19 AM, Ben Bacarisse wrote:
> Nam Nguyen <namducnguyen@shaw.ca> writes:
> <snip>

>>>>> Ben Bacarisse wrote: (quoting Nam)
> <snip>
>>>>>> Def-03a: even1(x) <-> Ey[x=y+y]
>>>>>> Def-03b: even2(x) <-> Ey[x=2*y]
>>>>>> Def-03c: even(x) <-> (even1(x) \/ even2(x))

> <snip>
>> ... _an odd number can not be defined without addition_ while
>> an even number can (as per Def-03b above).

>
> odd(x) <-> ~Ey[x=2*y]


Yes. There's something though from my original post (link) that has
missed being mentioned here but is _quite relevant_ . In that post
I had:

'Def-01: A formula is "positively assertive", or just "positive", iff
the formula contains no negation sign '~' ...'

So my answer to your question below is:
>
> In what sense does your definition of even2 avoid addition, where this
> one of odd does not?


in the sense that the formula defining even2 is a _positive_ formula
while the formula for your odd(x) above is _not_ .
>
> Personally, I'd say that both this and Def-03b use addition since
> multiplication in PA is usually defined using addition, but my
> view of what "without addition" means is not the issue here.


So then with the caveat above about the requirement of positive formula
here, can you define odd(x), without addition *and* using _only_
positive formula?

If you can't, then my comments about "overly complex" and Induction-
non-Induction duality definitions of even(x) but _not_ of odd(x), and
the implication thereof in my proof still stands.

--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI


Date Subject Author
8/12/13
Read Re: A finite set of all naturals
Ben Bacarisse
8/12/13
Read Re: A finite set of all naturals
Peter Percival
8/12/13
Read Re: A finite set of all naturals
fom
8/12/13
Read Re: A finite set of all naturals
Ben Bacarisse
8/12/13
Read Re: A finite set of all naturals
namducnguyen
8/12/13
Read Re: A finite set of all naturals
namducnguyen
8/12/13
Read Re: A finite set of all naturals
antani
8/12/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
Marshall
8/13/13
Read Re: A finite set of all naturals
quasi
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
quasi
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
quasi
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/14/13
Read Re: A finite set of all naturals
quasi
8/14/13
Read Re: A finite set of all naturals
namducnguyen
8/14/13
Read Re: A finite set of all naturals
quasi
8/14/13
Read Re: A finite set of all naturals
namducnguyen
8/15/13
Read Re: A finite set of all naturals
namducnguyen
8/15/13
Read Re: A finite set of all naturals
Virgil
8/15/13
Read Re: A finite set of all naturals
namducnguyen
8/15/13
Read Re: A finite set of all naturals
antani
8/13/13
Read Re: A finite set of all naturals
antani
8/13/13
Read Re: A finite set of all naturals
Peter Percival
8/13/13
Read Re: A finite set of all naturals
Ben Bacarisse
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/14/13
Read Re: A finite set of all naturals
Peter Percival
8/14/13
Read Re: A finite set of all naturals
Shmuel (Seymour J.) Metz

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