Re: Are there real numbers that cannot be defined?

> On 10 Mar 2019, at 13:52, Philip Thrift wrote: > > > > On Sunday, March 10, 2019 at 7:10:40 AM UTC-5, Bruno Marchal wrote: > >> On 8 Mar 2019, at 11:16, Philip Thrift > >> wrote: >> >> >> >> On Friday, March 8, 2019 at 3:18:39 AM UTC-6, Bruno Marchal wrote: >> >>> On 7 Mar 2019, at

Re: Are there real numbers that cannot be defined?

> On 8 Mar 2019, at 11:16, Philip Thrift wrote: > > > > On Friday, March 8, 2019 at 3:18:39 AM UTC-6, Bruno Marchal wrote: > >> On 7 Mar 2019, at 12:26, Philip Thrift > >> wrote: >> >> >> >> On Thursday, March 7, 2019 at 5:11:57 AM UTC-6, Bruno Marchal wrote: >> >>> On 6 Mar 2019, at

Re: Are there real numbers that cannot be defined?

On Friday, March 8, 2019 at 3:18:39 AM UTC-6, Bruno Marchal wrote: > > > On 7 Mar 2019, at 12:26, Philip Thrift > > wrote: > > > > On Thursday, March 7, 2019 at 5:11:57 AM UTC-6, Bruno Marchal wrote: >> >> >> On 6 Mar 2019, at 14:43, John Clark wrote: >> >> On Wed, Mar 6, 2019 at 8:30 AM Bruno

Re: Are there real numbers that cannot be defined?

> On 7 Mar 2019, at 12:26, Philip Thrift wrote: > > > > On Thursday, March 7, 2019 at 5:11:57 AM UTC-6, Bruno Marchal wrote: > >> On 6 Mar 2019, at 14:43, John Clark > wrote: >> >> On Wed, Mar 6, 2019 at 8:30 AM Bruno Marchal > > wrote: >> >> > You confirm my theory that strong (non

Re: Are there real numbers that cannot be defined?

> On 6 Mar 2019, at 14:43, John Clark wrote: > > On Wed, Mar 6, 2019 at 8:30 AM Bruno Marchal > wrote: > > > You confirm my theory that strong (non agnostic) atheism is radical > > religious fundamentalism > > I've never heard you or anybody else criticize me that

Re: Are there real numbers that cannot be defined?

On Wed, Mar 6, 2019 at 8:30 AM Bruno Marchal wrote: *> You confirm my theory that strong (non agnostic) atheism is radical > religious fundamentalism* I've never heard you or anybody else criticize me that brilliantly before, you sure put me in my place. I am devastated! > By theology, you

Re: Are there real numbers that cannot be defined?

> On 5 Mar 2019, at 19:13, John Clark wrote: > > > > On Tue, Mar 5, 2019 at 9:57 AM Bruno Marchal > wrote: > > > But in the “theology of the machine” [...] > > Given the fact that I don't have an infinite amount of time to read things my > rule of thumb is to

Re: Are there real numbers that cannot be defined?

> On 5 Mar 2019, at 15:53, John Clark wrote: > > > On Tue, Mar 5, 2019 at 8:03 AM Bruno Marchal > wrote: > > > The expression "Non computable numbers” appears only in intuitionist logic, > > If so then just by reading the title of Turing's famous 1936 paper where

Re: Are there real numbers that cannot be defined?

On Monday, March 4, 2019 at 3:35:55 AM UTC-7, Lawrence Crowell wrote: > > On Saturday, March 2, 2019 at 8:28:01 PM UTC-6, John Clark wrote: >> >> >> On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell >> wrote: >> >> > There are numbers that have no description in a practical sense. The >>>

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 9:57 AM Bruno Marchal wrote: *> But in the “theology of the machine”* [...] Given the fact that I don't have an infinite amount of time to read things my rule of thumb is to stop reading whenever I encounter the T word. John K Clark > -- You received this

Re: Are there real numbers that cannot be defined?

> On 5 Mar 2019, at 15:40, John Clark wrote: > > On Tue, Mar 5, 2019 at 7:40 AM Bruno Marchal > wrote: > > > (And what is proof anyway?) I did not wrote that. > > A proof is a construction made from a finite set of axioms using a finite set > of rules. Yes.

Re: Are there real numbers that cannot be defined?

> On 5 Mar 2019, at 15:20, John Clark wrote: > > On Tue, Mar 5, 2019 at 12:55 AM Russell Standish > wrote: > > > The usual meaning of computable integer is that there exists a program that > > outputs it. > > There is no point in arguing over the meaning of a

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 8:03 AM Bruno Marchal wrote: > *The expression "Non computable numbers” appears only in intuitionist > logic,* If so then just by reading the title of Turing's famous 1936 paper where he first described a device that we now call a Turing Machine you'd have to conclude

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 7:40 AM Bruno Marchal wrote: > *(And what is proof anyway?)* > A proof is a construction made from a finite set of axioms using a finite set of rules. If the axioms and the rules are sound then the proof will tell you something about the nature of reality, if they are not

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 12:55 AM Russell Standish wrote: > *The usual meaning of computable integer is that there exists a program > that outputs it.* There is no point in arguing over the meaning of a word, but if that is what you mean then there is a particular form of "computation" that is

Re: Are there real numbers that cannot be defined?

> On 5 Mar 2019, at 00:42, Bruce Kellett wrote: > > On Tue, Mar 5, 2019 at 10:25 AM Russell Standish > wrote: > On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote: > > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal > > wrote: > >

Re: Are there real numbers that cannot be defined?

> On 4 Mar 2019, at 23:31, John Clark wrote: > > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal > wrote: > > >> I don't follow you. If the 8000th BB number is unknowable then it is > >> certainly uncomputable > > > That is not true. All natural number n are

Re: Are there real numbers that cannot be defined?

> On 4 Mar 2019, at 21:34, Philip Thrift wrote: > > > > On Monday, March 4, 2019 at 12:00:05 PM UTC-6, John Clark wrote: > > > And proof is not truth. > ... > > John K Clark > > > > > Of course truth == proof in the land of radical intuitionists-constructivists. And John does not

Re: Are there real numbers that cannot be defined?

On Tuesday, March 5, 2019 at 12:23:28 AM UTC-6, Bruce wrote: > > On Tue, Mar 5, 2019 at 4:55 PM Russell Standish > wrote: > >> On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote: >> > On Tue, Mar 5, 2019 at 2:03 PM Russell Standish > > wrote: >> > >> > You cannot represent n as

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 4:55 PM Russell Standish wrote: > On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote: > > On Tue, Mar 5, 2019 at 2:03 PM Russell Standish > wrote: > > > > You cannot represent n as a finite string for an arbitrary real > number > > n. But you can for an

Re: Are there real numbers that cannot be defined?

On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote: > On Tue, Mar 5, 2019 at 2:03 PM Russell Standish wrote: > > On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote: > > > > My problem with your idea that the function: "(n-1)+1" is a valid > computational >

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 2:03 PM Russell Standish wrote: > On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote: > > > > My problem with your idea that the function: "(n-1)+1" is a valid > computational > > algorithm for n is that it makes all real numbers also computable, but > the > >

Re: Are there real numbers that cannot be defined?

On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote: > > My problem with your idea that the function: "(n-1)+1" is a valid > computational > algorithm for n is that it makes all real numbers also computable, but the > notion of Turing computability applies only to the integers. We do

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 11:50 AM Russell Standish wrote: > On Tue, Mar 05, 2019 at 10:42:00AM +1100, Bruce Kellett wrote: > > On Tue, Mar 5, 2019 at 10:25 AM Russell Standish > wrote: > > > > On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote: > > > On Mon, Mar 4, 2019 at 11:04

Re: Are there real numbers that cannot be defined?

On Tue, Mar 05, 2019 at 10:42:00AM +1100, Bruce Kellett wrote: > On Tue, Mar 5, 2019 at 10:25 AM Russell Standish > wrote: > > On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote: > > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal wrote: > > > > > >         >> I don't

Re: Are there real numbers that cannot be defined?

On Mon, Mar 04, 2019 at 06:48:19PM -0500, John Clark wrote: > > On Mon, Mar 4, 2019 at 6:25 PM Russell Standish wrote: >   > > > OK, so what about the program "print X+1", where X is the expansion of > the number BB(8000)-1? > > > Well what about it? If you don't know what BB(8000)

Re: Are there real numbers that cannot be defined?

On Mon, Mar 4, 2019 at 6:25 PM Russell Standish wrote: > > > *OK, so what about the program "print X+1", where X is the expansion of > the number BB(8000)-1?* > Well what about it? If you don't know what BB(8000) is, and you don't and neither does God, then neither of you will ever know what

Re: Are there real numbers that cannot be defined?

On Tue, Mar 5, 2019 at 10:25 AM Russell Standish wrote: > On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote: > > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal wrote: > > > > > > >> I don't follow you. If the 8000th BB number is unknowable > then it is > > certainly

Re: Are there real numbers that cannot be defined?

On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote: > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal wrote: > > > >> I don't follow you. If the 8000th BB number is unknowable then it > is > certainly uncomputable > > > > That is not true. All natural number n are

Re: Are there real numbers that cannot be defined?

On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal wrote: >> I don't follow you. If the 8000th BB number is unknowable then it is >> certainly uncomputable > > > *> That is not true. All natural number n are computable. The program is > “output n”.* > I think you're being silly. You're saying if you

Re: Are there real numbers that cannot be defined?

On Monday, March 4, 2019 at 12:00:05 PM UTC-6, John Clark wrote: > > > > And proof is not truth. > ... > > John K Clark > > > >> Of course *truth == proof *in the land of radical intuitionists-constructivists. (And what is proof anyway?) From: Doren Zeilberger To: Scott Aaronson [

Re: Are there real numbers that cannot be defined?

On Mon, Mar 4, 2019 at 7:56 AM Bruno Marchal wrote: > *> BB(7918) is just a number, say k, and the program “print k” will do.* > Unless God wrote the program the output would just be "k" or perhaps "BB(7918)". And as I said before unlike most Real Numbers Busy Beaver Numbers can be named but

Re: Are there real numbers that cannot be defined?

> On 4 Mar 2019, at 14:24, John Clark wrote: > > On Sun, Mar 3, 2019 at 6:38 PM Russell Standish > wrote: > > > ISTM that the 8000th BB number is unknowable, rather than uncomputable. > > I don't follow you. If the 8000th BB number is unknowable then it is >

Re: Are there real numbers that cannot be defined?

> On 4 Mar 2019, at 11:35, Lawrence Crowell > wrote: > > On Saturday, March 2, 2019 at 8:28:01 PM UTC-6, John Clark wrote: > > On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell > wrote: > > > There are numbers that have no description in a practical sense. The > > numbers 10^{10^{10^{10}}}

Re: Are there real numbers that cannot be defined?

> On 4 Mar 2019, at 00:37, Russell Standish wrote: > > ISTM that the 8000th BB number is unknowable, rather than > uncomputable. As Bruno said, there is a program that outputs the > 8000th BB number, but we can never know that this program is the > correct one. Yes. Clark mention an

Re: Are there real numbers that cannot be defined?

> On 3 Mar 2019, at 22:58, Brent Meeker wrote: > > > > On 3/3/2019 1:37 PM, John Clark wrote: >> On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift > > wrote: >> >> > If a program "represents" a real number (e.g. in the spigot sense), then >> > that could be said to

Re: Are there real numbers that cannot be defined?

On Sun, Mar 3, 2019 at 6:38 PM Russell Standish wrote: *> ISTM that the 8000th BB number is unknowable, rather than uncomputable.* I don't follow you. If the 8000th BB number is unknowable then it is certainly uncomputable but if it's uncomputable then it's only *probably* unknowable because

Re: Are there real numbers that cannot be defined?

> On 3 Mar 2019, at 20:52, Philip Thrift wrote: > > > > On Sunday, March 3, 2019 at 11:29:32 AM UTC-6, John Clark wrote: > > > On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal > > wrote: >> >> >> The 8000th Busy Beaver Number can be named but not calculated even >> >> theoretically, > > The

Re: Are there real numbers that cannot be defined?

> On 3 Mar 2019, at 18:28, John Clark wrote: > > > > On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal > wrote: >> >> >> The 8000th Busy Beaver Number can be named but not calculated even >> >> theoretically, > > The busy beaver function is not computable, but on each

Re: Are there real numbers that cannot be defined?

On Saturday, March 2, 2019 at 8:28:01 PM UTC-6, John Clark wrote: > > > On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell > wrote: > > > There are numbers that have no description in a practical sense. The >> numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a vast number of >> numbers

Re: Are there real numbers that cannot be defined?

ISTM that the 8000th BB number is unknowable, rather than uncomputable. As Bruno said, there is a program that outputs the 8000th BB number, but we can never know that this program is the correct one. Cheers On Sun, Mar 03, 2019 at 12:28:54PM -0500, John Clark wrote: > > > On Sun, Mar 3, 2019

Re: Are there real numbers that cannot be defined?

On Mon, Mar 4, 2019 at 9:13 AM Bruce Kellett wrote: > On Mon, Mar 4, 2019 at 8:58 AM Brent Meeker wrote: > >> On 3/3/2019 1:37 PM, John Clark wrote: >> >> On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift >> wrote: >> >> *> If a program "represents" a real number (e.g. in the spigot sense), >>>

Re: Are there real numbers that cannot be defined?

On Mon, Mar 4, 2019 at 8:58 AM Brent Meeker wrote: > On 3/3/2019 1:37 PM, John Clark wrote: > > On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift > wrote: > > *> If a program "represents" a real number (e.g. in the spigot sense), >> then that could be said to "define" it.* > > > But for most Real

Re: Are there real numbers that cannot be defined?

On Sunday, March 3, 2019 at 3:38:18 PM UTC-6, John Clark wrote: > > On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift > wrote: > > *> If a program "represents" a real number (e.g. in the spigot sense), >> then that could be said to "define" it.* > > > But for most Real Numbers there is no such

Re: Are there real numbers that cannot be defined?

On 3/3/2019 1:37 PM, John Clark wrote: On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift > wrote: /> If a program "represents" a real number (e.g. in the spigot sense), then that could be said to "define" it./ But for most Real Numbers there is no such

Re: Are there real numbers that cannot be defined?

On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift wrote: *> If a program "represents" a real number (e.g. in the spigot sense), then > that could be said to "define" it.* But for most Real Numbers there is no such program. * > But what does it mean for a real number to be "defined"?* If you can

Re: Are there real numbers that cannot be defined?

On Sunday, March 3, 2019 at 11:29:32 AM UTC-6, John Clark wrote: > > > > On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal > wrote: > >> >> >> The 8000th Busy Beaver Number can be named but not calculated even >>> theoretically, >> >> >> *> The busy beaver function is not computable, but on each

Re: Are there real numbers that cannot be defined?

On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal wrote: > > >> The 8000th Busy Beaver Number can be named but not calculated even >> theoretically, > > > *> The busy beaver function is not computable, but on each individual n, > it is computable theoretically, * > No it is not, not if n= 7918, to

Re: Are there real numbers that cannot be defined?

On Sun, Mar 3, 2019 at 3:25 AM Philip Thrift wrote: The 8000th Busy Beaver Number can be named but not calculated even >> theoretically, but most Real Numbers can't even be uniquely named with >> ASCII characters, not even with an infinite number of them. >> >> > > *The "computable" real numbers

Re: Are there real numbers that cannot be defined?

> On 3 Mar 2019, at 03:27, John Clark wrote: > > > On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell > mailto:goldenfieldquaterni...@gmail.com>> > wrote: > > > There are numbers that have no description in a practical sense. The > > numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a

Re: Are there real numbers that cannot be defined?

On Saturday, March 2, 2019 at 8:28:01 PM UTC-6, John Clark wrote: > > > On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell > wrote: > > > There are numbers that have no description in a practical sense. The >> numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a vast number of >> numbers

Re: Are there real numbers that cannot be defined?

On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell < goldenfieldquaterni...@gmail.com> wrote: > There are numbers that have no description in a practical sense. The > numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a vast number of > numbers that have no description with any information

Re: Are there real numbers that cannot be defined?

On Friday, March 1, 2019 at 7:02:20 PM UTC-6, Lawrence Crowell wrote: > > On Friday, March 1, 2019 at 4:13:00 PM UTC-6, Brent wrote: >> >> >> >> On 3/1/2019 1:44 PM, Philip Thrift wrote: >> >> >> >> I think the interesting question is what does the function on natural >> numbers >> >> n

Re: Are there real numbers that cannot be defined?

On Friday, March 1, 2019 at 4:13:00 PM UTC-6, Brent wrote: > > > > On 3/1/2019 1:44 PM, Philip Thrift wrote: > > > > I think the interesting question is what does the function on natural > numbers > > n → shortestDescriptionOf(n)/n > > look like. > > > That's not even a function since

Re: Are there real numbers that cannot be defined?

On 3/1/2019 1:44 PM, Philip Thrift wrote: I think the interesting question is what does the function on natural numbers       n → shortestDescriptionOf(n)/n look like. That's not even a function since ShortestDescriptionOf (n) depends on the notation for the description and different

Re: Are there real numbers that cannot be defined?

I think the interesting question is what does the function on natural numbers n → shortestDescriptionOf(n)/n look like. - pt On Friday, March 1, 2019 at 3:22:59 PM UTC-6, Lawrence Crowell wrote: > > There are numbers that have no description in a practical sense. The > numbers

Re: Are there real numbers that cannot be defined?

There are numbers that have no description in a practical sense. The numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a vast number of numbers that have no description with any information theoretic sense. This is even if all the particles in the some 10^{500}cosmologies in the

Re: Are there real numbers that cannot be defined?

On Thursday, February 28, 2019 at 10:54:28 AM UTC-6, Bruno Marchal wrote: > > > On 27 Feb 2019, at 19:18, Philip Thrift > > wrote: > > > > On Wednesday, February 27, 2019 at 11:25:06 AM UTC-6, Bruno Marchal wrote: >> >> >> On 26 Feb 2019, at 19:41, Philip Thrift wrote: >> >> >> >> There is

Re: Are there real numbers that cannot be defined?

> On 27 Feb 2019, at 19:18, Philip Thrift wrote: > > > > On Wednesday, February 27, 2019 at 11:25:06 AM UTC-6, Bruno Marchal wrote: > >> On 26 Feb 2019, at 19:41, Philip Thrift > >> wrote: >> >> >> >> On Tuesday, February 26, 2019 at 5:07:38 AM UTC-6, Bruno Marchal wrote: >> >>> On 25

Re: Are there real numbers that cannot be defined?

On Wednesday, February 27, 2019 at 11:25:06 AM UTC-6, Bruno Marchal wrote: > > > On 26 Feb 2019, at 19:41, Philip Thrift > > wrote: > > > > On Tuesday, February 26, 2019 at 5:07:38 AM UTC-6, Bruno Marchal wrote: >> >> >> On 25 Feb 2019, at 20:35, Philip Thrift wrote: >> >> via >>

Re: Are there real numbers that cannot be defined?

> On 26 Feb 2019, at 19:41, Philip Thrift wrote: > > > > On Tuesday, February 26, 2019 at 5:07:38 AM UTC-6, Bruno Marchal wrote: > >> On 25 Feb 2019, at 20:35, Philip Thrift > >> wrote: >> >> via >> https://twitter.com/JDHamkins/status/1100090709527408640 >>

Re: Are there real numbers that cannot be defined?

On Tuesday, February 26, 2019 at 5:07:38 AM UTC-6, Bruno Marchal wrote: > > > On 25 Feb 2019, at 20:35, Philip Thrift > > wrote: > > via > https://twitter.com/JDHamkins/status/1100090709527408640 > > Joel David Hamkins @JDHamkins > > *Must there be numbers we cannot describe or define?

Re: Are there real numbers that cannot be defined?

> On 25 Feb 2019, at 20:35, Philip Thrift wrote: > > via > https://twitter.com/JDHamkins/status/1100090709527408640 > > Joel David Hamkins @JDHamkins > > Must there be numbers we cannot describe or define? Definability in > mathematics and the Math Tea argument > Pure Mathematics Research

Are there real numbers that cannot be defined?

via https://twitter.com/JDHamkins/status/1100090709527408640 Joel David Hamkins @JDHamkins *Must there be numbers we cannot describe or define? Definability in mathematics and the Math Tea argument* Pure Mathematics Research Seminar at the University of East Anglia in Norwich on Monday, 25