Re: The Axiom Of Choice and ComputationalismT

2015-09-28 Thread John Clark
On Mon, Sep 28, 2015 Bruno Marchal wrote: t > ​> ​ > he same set of axiom, ZF, can have a model verifying AC and a model > verifying ~AC, > ​Yes, we've known that since 1963 and therefore AC is independent of ZF. ​ > ​>>​ >> ​emulated people have access to arithmetic just

Re: The Axiom Of Choice and ComputationalismT

2015-09-27 Thread John Clark
On Sun, Sep 27, 2015 at 12:04 PM, Bruno Marchal wrote: ​> ​ > The constructible set of Gödel can be use to show that ZF and ZFC proves > the same arithmetical theorems. ​That is incorrect, ZF can not prove that the Banach-Tarski construction works, ZFC can. What Godel

Re: The Axiom Of Choice and ComputationalismT

2015-09-26 Thread Bruno Marchal
On 25 Sep 2015, at 19:16, John Clark wrote: On Fri, Sep 25, 2015 at 11:36 AM, Bruno Marchal wrote: ​>>​Paul Cohen not Godel proved that arithmetical reality is independent of the​ ​Axiom of Choice ​> ​I don't think so. The independence of arithmetic from AC in

Re: The Axiom Of Choice and ComputationalismT

2015-09-26 Thread John Clark
On Sat, Sep 26, 2015 Bruno Marchal wrote: > >>> ​>>​ >>> Paul Cohen not Godel proved that arithmetical reality is independent of >>> the >>> ​ ​ >>> Axiom of Choice >> >> >> > ​> ​ >> I don't think so. The independence of arithmetic from AC in ZF follows >> from Gödel's proof

Re: The Axiom Of Choice and ComputationalismT

2015-09-25 Thread Bruno Marchal
On 24 Sep 2015, at 20:49, John Clark wrote: On Thu, Sep 24, 2015 at Bruno Marchal wrote: ​> ​You can define prime number in arithmetic, ​Who cares? I'm not interested in ​arithmetic or in anything else defining prime numbers, I'm interested in CALCULATING prime

Re: The Axiom Of Choice and ComputationalismT

2015-09-25 Thread John Clark
On Fri, Sep 25, 2015 at 11:36 AM, Bruno Marchal wrote: >> ​>>​ >> Paul Cohen not Godel proved that arithmetical reality is independent of >> the >> ​ ​ >> Axiom of Choice > > > ​> ​ > I don't think so. The independence of arithmetic from AC in ZF follows > from Gödel's proof

Re: The Axiom Of Choice and ComputationalismT

2015-09-24 Thread Bruno Marchal
On 23 Sep 2015, at 23:59, John Clark wrote: It seems to me the debate I’v been having with Bruno, the one about Arithmetic being able to perform calculations all by itself without the help of matter that obeys the laws of physics, comes down to the Axiom Of Choice. I would humbly propose

Re: The Axiom Of Choice and ComputationalismT

2015-09-24 Thread John Clark
On Thu, Sep 24, 2015 at Bruno Marchal wrote: ​> ​ > You can define prime number in arithmetic, > ​Who cares? I'm not interested in ​arithmetic or in anything else defining prime numbers, I'm interested in *CALCULATING *prime numbers. ​> ​ > That the arithmetical reality is

The Axiom Of Choice and ComputationalismT

2015-09-23 Thread John Clark
It seems to me the debate I’v been having with Bruno, the one about Arithmetic being able to perform calculations all by itself without the help of matter that obeys the laws of physics, comes down to the Axiom Of Choice. I would humbly propose that maybe just maybe mathematics is everything