There is no possibility of introducing something absolutely uncountable, but by a pure dogma. [T. Skolem: "Über einige Grundlagenfragen der Mathematik", Norske Videnskaps-Akademi, Mat.-naturv. Klasse No. 4, Oslo (1929)]

In order to obtain something absolutely nondenumerable, we would have to have either an absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely nondenumerable number of first-order propositions. [...] It is easy to show that Zermelo's axiom system is not sufficient to provide a complete foundation for the usual theory of sets. [p. 296] I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique." [p. 300f] [T. Skolem: "Some remarks on axiomatized set theory" (1922) quoted in J. van Heijenoort: "From Frege to Gödel – A source book in mathematical logic, 1879-1931", Harvard University Press, Cambridge (1967)]

Of course we know that, e.g., the Axiom of Choice is required for many mathematical theorems (such as: every vector space has a basis), which in turn can be applied in physics. However, on closer inspection it turns out that for all concrete instances that are used, the axiom of choice is not required. The same applies even for the existence of an infinite set: One can use a very constructive, "finitary" form of mathematics that is perfectly sufficient for physics. [...] the old intuition: "if nontrivial set theory, non-constructive mathematics or a non-measurable set is used in an essential way, it cannot be physically relevant". [J. Kellner: "Pitowsky's Kolmogorovian models and super-determinism", arXiv (2016) p. 11f]

Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]

Regards, WM

torsdag 2 december 2021 kl. 09:50:41 UTC+1 skrev WM:
> There is no possibility of introducing something absolutely uncountable, but by a pure dogma. [T. Skolem: "Über einige Grundlagenfragen der Mathematik", Norske Videnskaps-Akademi, Mat.-naturv. Klasse No. 4, Oslo (1929)]
>
> In order to obtain something absolutely nondenumerable, we would have to have either an absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely nondenumerable number of first-order propositions. [...] It is easy to show that Zermelo's axiom system is not sufficient to provide a complete foundation for the usual theory of sets. [p. 296] I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique." [p. 300f] [T. Skolem: "Some remarks on axiomatized set theory" (1922) quoted in J. van Heijenoort: "From Frege to Gödel – A source book in mathematical logic, 1879-1931", Harvard University Press, Cambridge (1967)]
>
> Of course we know that, e.g., the Axiom of Choice is required for many mathematical theorems (such as: every vector space has a basis), which in turn can be applied in physics. However, on closer inspection it turns out that for all concrete instances that are used, the axiom of choice is not required. The same applies even for the existence of an infinite set: One can use a very constructive, "finitary" form of mathematics that is perfectly sufficient for physics. [...] the old intuition: "if nontrivial set theory, non-constructive mathematics or a non-measurable set is used in an essential way, it cannot be physically relevant". [J. Kellner: "Pitowsky's Kolmogorovian models and super-determinism", arXiv (2016) p. 11f]
>
> Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
>
> Regards, WM

are you ever going to produce anything meaningful?

On Thursday, 2 December 2021 at 06:39:12 UTC-4, zelos...@gmail.com wrote:
> torsdag 2 december 2021 kl. 09:50:41 UTC+1 skrev WM:

> > Regards, WM
> are you ever going to produce anything meaningful?

That's rhetorical, right? He hasn't produced anything meaningful, or even correct, since at least 2005, and the decline of his mental facilities is at this point painfully obvious for everybody.

torsdag 2 december 2021 kl. 13:09:09 UTC+1 skrev Gus Gassmann:
> On Thursday, 2 December 2021 at 06:39:12 UTC-4, zelos...@gmail.com wrote:
> > torsdag 2 december 2021 kl. 09:50:41 UTC+1 skrev WM:
>
> > > Regards, WM
> > are you ever going to produce anything meaningful?
> That's rhetorical, right? He hasn't produced anything meaningful, or even correct, since at least 2005, and the decline of his mental facilities is at this point painfully obvious for everybody.
yepp, but still.

T. Skolem did not say "and useless" You added that.

On 12/2/2021 2:50 AM, WM wrote:
> There is no possibility of introducing something absolutely uncountable, but by a pure dogma. [T. Skolem: "Über einige Grundlagenfragen der Mathematik", Norske Videnskaps-Akademi, Mat.-naturv. Klasse No. 4, Oslo (1929)]

your dark numbers are uncountable, so they are not possible either.

>
> Regards, WM
>

On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> [...]
> Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]

It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.

On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
> On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> > [...]
> > Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
> It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.

Ummm.... guys, I am in danger of unifying the integer with the continuum. As your numbers turn large do they care about the little end of things? No, they do not. The large values outweight the littler digits. Like siblings the biggest get first dibs, and but for their kindness and respect to the little ones there is not much going for them. Of course we would like an egalitarian number system. But when would you say it is true that one million versus one million and one mark a breaking point? In some base threshing mechanism perhaps, and in that mechanism I might well ask whether two million versus two million and one mark some stark breakpoint? Well then, you've been engaging in radix principles, perhaps.

Interesting possibility: if you did claim to care about sin(x) in N and create some variants on this would ye be recovering a radix system? Why, you'd have your lower digits in no time. Did you want to radix three? Is it any mystery where the next mark is? But for labels these are reusables and they are asking for something that is foreign to your system. Inducing rotational qualities into your values is this easy. Maybe I'll try to expand this out a bit, but clearly we will see that
sin( 3 ), sin( 4 ), sin( 5 ), sin( 6 ), sin( 7 ), ...
beg that we put a discrete toothed ratchet in, doesn't it?
I suppose you could try to go with something else, but it won't be very n-ary. But you see you can't have the exploits of the real values system already. two pi is not a thing here yet, so let's just round down to six and we'll call this beast a dirty re-radixer.
So you see that when you plug your dirty re-radixer into another dirty re-radixer that you get patterns? Well, maybe I am wrong to be restrictive and insist on an unmixed radix. Mix it up; all the better; shall we let the next one be a radix seven? then the prior was a five? Then this would put the six out at 6!, no? 120... now that is handy. Could you imagine the poor superstitious cultures that would accrue from such a system? Is 360 so far away? Then too, back at two... a sticking point for humans it seems, the first signs of NU and MU as unique appear. And of course you who are N-ary are none the wiser, but for a little dot.

Timothy Golden presented the following explanation :
> On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
>> On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
>>> [...]
>>> Objects whose existence is postulated by this axiom {{Zermelo's axiom of
>>> choice}} appeared to be not only useless but sometimes destructive to the
>>> simplicity and rigorousness of crucial mathematical theories. [A.N.
>>> Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6
>>> (1929) pp. 41-54]
>> It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the
>> journal properly any longer. Note also that Kolmogorov uses "appeared" (note
>> even "appear", but past tense) in an article on "debates". He also mentions
>> Hilbert's approach, but of course you chose to ignore that entirely. Your
>> selective quotations, taken out of context, are unethical and
>> unprofessional. But we knew that you are a lying, cheating son of a bitch,
>> so, nothing to see here.
>
> Ummm.... guys, I am in danger of unifying the integer with the continuum

Not really.

> as your numbers turn large do they care about the little end of things? No,
> they do not.

Yes they do, in fact calculations of residue often use modular
arithmetic and discard the integer part completely so that numbers
don't accumulate large values. Cryptography uses modular arithmetic a
lot.

On Thursday, December 2, 2021 at 3:18:34 PM UTC+1, Gus Gassmann wrote:
> On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> > [...]
> > Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
> >
> It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.

WM: pure dogma - and useless

On Thursday, December 2, 2021 at 11:04:20 AM UTC-5, FromTheRafters wrote:
> Timothy Golden presented the following explanation :
> > On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
> >> On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> >>> [...]
> >>> Objects whose existence is postulated by this axiom {{Zermelo's axiom of
> >>> choice}} appeared to be not only useless but sometimes destructive to the
> >>> simplicity and rigorousness of crucial mathematical theories. [A.N.
> >>> Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6
> >>> (1929) pp. 41-54]
> >> It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the
> >> journal properly any longer. Note also that Kolmogorov uses "appeared" (note
> >> even "appear", but past tense) in an article on "debates". He also mentions
> >> Hilbert's approach, but of course you chose to ignore that entirely. Your
> >> selective quotations, taken out of context, are unethical and
> >> unprofessional. But we knew that you are a lying, cheating son of a bitch,
> >> so, nothing to see here.
> >
> > Ummm.... guys, I am in danger of unifying the integer with the continuum
> Not really.
Yes, really:
1.23 = 123 with a two for the secondary unity!

>
> > as your numbers turn large do they care about the little end of things? No,
> > they do not.
>
> Yes they do, in fact calculations of residue often use modular
> arithmetic and discard the integer part completely so that numbers
> don't accumulate large values. Cryptography uses modular arithmetic a
> lot.

Those that say math isn't need of ZFC etc.. are especially
those that are in need of ZFC etc.. Examples:

- WM: Famous example Mückenschuss inference rule.

- Dan-O-Matik: Just WM with extra steps, cannot
read and understand a single formula, and fails

in beginner math. He still struggles with an example
that can be derived from axiom schema of replacement.

- What else?

WM schrieb am Donnerstag, 2. Dezember 2021 um 09:50:41 UTC+1:
> There is no possibility of introducing something absolutely uncountable, but by a pure dogma. [T. Skolem: "Über einige Grundlagenfragen der Mathematik", Norske Videnskaps-Akademi, Mat.-naturv. Klasse No. 4, Oslo (1929)]
>
> In order to obtain something absolutely nondenumerable, we would have to have either an absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely nondenumerable number of first-order propositions. [...] It is easy to show that Zermelo's axiom system is not sufficient to provide a complete foundation for the usual theory of sets. [p. 296] I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique." [p. 300f] [T. Skolem: "Some remarks on axiomatized set theory" (1922) quoted in J. van Heijenoort: "From Frege to Gödel – A source book in mathematical logic, 1879-1931", Harvard University Press, Cambridge (1967)]
>
> Of course we know that, e.g., the Axiom of Choice is required for many mathematical theorems (such as: every vector space has a basis), which in turn can be applied in physics. However, on closer inspection it turns out that for all concrete instances that are used, the axiom of choice is not required. The same applies even for the existence of an infinite set: One can use a very constructive, "finitary" form of mathematics that is perfectly sufficient for physics. [...] the old intuition: "if nontrivial set theory, non-constructive mathematics or a non-measurable set is used in an essential way, it cannot be physically relevant". [J. Kellner: "Pitowsky's Kolmogorovian models and super-determinism", arXiv (2016) p. 11f]
>
> Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
>
> Regards, WM

On 12/2/2021 11:01 AM, Timothy Golden wrote:
> On Thursday, December 2, 2021 at 11:04:20 AM UTC-5, FromTheRafters wrote:
>> Timothy Golden presented the following explanation :
>>> On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
>>>> On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
>>>>> [...]
>>>>> Objects whose existence is postulated by this axiom {{Zermelo's axiom of
>>>>> choice}} appeared to be not only useless but sometimes destructive to the
>>>>> simplicity and rigorousness of crucial mathematical theories. [A.N.
>>>>> Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6
>>>>> (1929) pp. 41-54]
>>>> It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the
>>>> journal properly any longer. Note also that Kolmogorov uses "appeared" (note
>>>> even "appear", but past tense) in an article on "debates". He also mentions
>>>> Hilbert's approach, but of course you chose to ignore that entirely. Your
>>>> selective quotations, taken out of context, are unethical and
>>>> unprofessional. But we knew that you are a lying, cheating son of a bitch,
>>>> so, nothing to see here.
>>>
>>> Ummm.... guys, I am in danger of unifying the integer with the continuum
>> Not really.
> Yes, really:
> 1.23 = 123 with a two for the secondary unity!

1.23 does not equal 123.

>
>>
>>> as your numbers turn large do they care about the little end of things? No,
>>> they do not.
>>
>> Yes they do, in fact calculations of residue often use modular
>> arithmetic and discard the integer part completely so that numbers
>> don't accumulate large values. Cryptography uses modular arithmetic a
>> lot.

torsdag 2 december 2021 kl. 15:51:54 UTC+1 skrev timba...@gmail.com:
> On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
> > On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> > > [...]
> > > Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
> > It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.
> Ummm.... guys, I am in danger of unifying the integer with the continuum. As your numbers turn large do they care about the little end of things? No, they do not. The large values outweight the littler digits. Like siblings the biggest get first dibs, and but for their kindness and respect to the little ones there is not much going for them. Of course we would like an egalitarian number system. But when would you say it is true that one million versus one million and one mark a breaking point? In some base threshing mechanism perhaps, and in that mechanism I might well ask whether two million versus two million and one mark some stark breakpoint? Well then, you've been engaging in radix principles, perhaps.
>
> Interesting possibility: if you did claim to care about sin(x) in N and create some variants on this would ye be recovering a radix system? Why, you'd have your lower digits in no time. Did you want to radix three? Is it any mystery where the next mark is? But for labels these are reusables and they are asking for something that is foreign to your system. Inducing rotational qualities into your values is this easy. Maybe I'll try to expand this out a bit, but clearly we will see that
> sin( 3 ), sin( 4 ), sin( 5 ), sin( 6 ), sin( 7 ), ...
> beg that we put a discrete toothed ratchet in, doesn't it?
> I suppose you could try to go with something else, but it won't be very n-ary. But you see you can't have the exploits of the real values system already. two pi is not a thing here yet, so let's just round down to six and we'll call this beast a dirty re-radixer.
> So you see that when you plug your dirty re-radixer into another dirty re-radixer that you get patterns? Well, maybe I am wrong to be restrictive and insist on an unmixed radix. Mix it up; all the better; shall we let the next one be a radix seven? then the prior was a five? Then this would put the six out at 6!, no? 120... now that is handy. Could you imagine the poor superstitious cultures that would accrue from such a system? Is 360 so far away? Then too, back at two... a sticking point for humans it seems, the first signs of NU and MU as unique appear. And of course you who are N-ary are none the wiser, but for a little dot.

are you going insane?

An the winner for going insane is Dan-O-Matik:

Dan-O-Matik halucinated his own little crazy world:
> Given s0={0}, f:s0-->s0 and ~1=0, we have: f(0)=0 and f(1) is indeterminate (undefined)
> Formal Proof: https://www.dcproof.com/FunctionUndefined.htm (23 lines)

This is not correct mathematics. If you say f : X -> Y, you
want the domain of f to be determined to be X.
If you have for f : {0} -> {0}, f(1) inderminate, then dom(f)

can be either {0} or {0,1}. You need to have f(1) undefined,
so that the domain is always {0}. You can check any standard
reference in mathematics, for example:

Basic Set Theory - Azriel Levy
https://www.amazon.de/dp/0486420795

The booklet is only 20 bucks or so.

zelos...@gmail.com schrieb am Freitag, 3. Dezember 2021 um 06:24:09 UTC+1:
> torsdag 2 december 2021 kl. 15:51:54 UTC+1 skrev timba...@gmail.com:
> > On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
> > > On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> > > > [...]
> > > > Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
> > > It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.
> > Ummm.... guys, I am in danger of unifying the integer with the continuum. As your numbers turn large do they care about the little end of things? No, they do not. The large values outweight the littler digits. Like siblings the biggest get first dibs, and but for their kindness and respect to the little ones there is not much going for them. Of course we would like an egalitarian number system. But when would you say it is true that one million versus one million and one mark a breaking point? In some base threshing mechanism perhaps, and in that mechanism I might well ask whether two million versus two million and one mark some stark breakpoint? Well then, you've been engaging in radix principles, perhaps.
> >
> > Interesting possibility: if you did claim to care about sin(x) in N and create some variants on this would ye be recovering a radix system? Why, you'd have your lower digits in no time. Did you want to radix three? Is it any mystery where the next mark is? But for labels these are reusables and they are asking for something that is foreign to your system. Inducing rotational qualities into your values is this easy. Maybe I'll try to expand this out a bit, but clearly we will see that
> > sin( 3 ), sin( 4 ), sin( 5 ), sin( 6 ), sin( 7 ), ...
> > beg that we put a discrete toothed ratchet in, doesn't it?
> > I suppose you could try to go with something else, but it won't be very n-ary. But you see you can't have the exploits of the real values system already. two pi is not a thing here yet, so let's just round down to six and we'll call this beast a dirty re-radixer.
> > So you see that when you plug your dirty re-radixer into another dirty re-radixer that you get patterns? Well, maybe I am wrong to be restrictive and insist on an unmixed radix. Mix it up; all the better; shall we let the next one be a radix seven? then the prior was a five? Then this would put the six out at 6!, no? 120... now that is handy. Could you imagine the poor superstitious cultures that would accrue from such a system? Is 360 so far away? Then too, back at two... a sticking point for humans it seems, the first signs of NU and MU as unique appear. And of course you who are N-ary are none the wiser, but for a little dot.
> are you going insane?

Ask Dan-O-Matik about cardinal arithmetic, he will not have
any clue. Another notation for the function space A -> B, is B^A.
And then we have this nice bijection theorem:

P(A) ~ 2^A

Subsequently:

|P(A)| = |2^A|

For example N is countable and 2^N is uncountable. Does Dan-O-Matik
even have the slightest idea what functions spaces are?

See also:
Given two sets S and T, the set of all functions from T to S is denoted S^T..
This fits in with the exponentiation of cardinal numbers, in the sense
that |S^T| = |S|^|T|, where |X| is the cardinality of X.
https://en.wikipedia.org/wiki/Exponentiation#Sets_as_exponents

Mostowski Collapse schrieb am Freitag, 3. Dezember 2021 um 09:40:54 UTC+1:
> An the winner for going insane is Dan-O-Matik:
>
> Dan-O-Matik halucinated his own little crazy world:
> > Given s0={0}, f:s0-->s0 and ~1=0, we have: f(0)=0 and f(1) is indeterminate (undefined)
> > Formal Proof: https://www.dcproof.com/FunctionUndefined.htm (23 lines)
>
> This is not correct mathematics. If you say f : X -> Y, you
> want the domain of f to be determined to be X.
> If you have for f : {0} -> {0}, f(1) inderminate, then dom(f)
>
> can be either {0} or {0,1}. You need to have f(1) undefined,
> so that the domain is always {0}. You can check any standard
> reference in mathematics, for example:
>
> Basic Set Theory - Azriel Levy
> https://www.amazon.de/dp/0486420795
>
> The booklet is only 20 bucks or so.
> zelos...@gmail.com schrieb am Freitag, 3. Dezember 2021 um 06:24:09 UTC+1:
> > torsdag 2 december 2021 kl. 15:51:54 UTC+1 skrev timba...@gmail.com:
> > > On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
> > > > On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> > > > > [...]
> > > > > Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
> > > > It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.
> > > Ummm.... guys, I am in danger of unifying the integer with the continuum. As your numbers turn large do they care about the little end of things? No, they do not. The large values outweight the littler digits. Like siblings the biggest get first dibs, and but for their kindness and respect to the little ones there is not much going for them. Of course we would like an egalitarian number system. But when would you say it is true that one million versus one million and one mark a breaking point? In some base threshing mechanism perhaps, and in that mechanism I might well ask whether two million versus two million and one mark some stark breakpoint? Well then, you've been engaging in radix principles, perhaps.
> > >
> > > Interesting possibility: if you did claim to care about sin(x) in N and create some variants on this would ye be recovering a radix system? Why, you'd have your lower digits in no time. Did you want to radix three? Is it any mystery where the next mark is? But for labels these are reusables and they are asking for something that is foreign to your system. Inducing rotational qualities into your values is this easy. Maybe I'll try to expand this out a bit, but clearly we will see that
> > > sin( 3 ), sin( 4 ), sin( 5 ), sin( 6 ), sin( 7 ), ...
> > > beg that we put a discrete toothed ratchet in, doesn't it?
> > > I suppose you could try to go with something else, but it won't be very n-ary. But you see you can't have the exploits of the real values system already. two pi is not a thing here yet, so let's just round down to six and we'll call this beast a dirty re-radixer.
> > > So you see that when you plug your dirty re-radixer into another dirty re-radixer that you get patterns? Well, maybe I am wrong to be restrictive and insist on an unmixed radix. Mix it up; all the better; shall we let the next one be a radix seven? then the prior was a five? Then this would put the six out at 6!, no? 120... now that is handy. Could you imagine the poor superstitious cultures that would accrue from such a system? Is 360 so far away? Then too, back at two... a sticking point for humans it seems, the first signs of NU and MU as unique appear. And of course you who are N-ary are none the wiser, but for a little dot.
> > are you going insane?

Dan-O-Matiks making fun of 3 ⊆ 5, and the same time not
being able to define when a two place predicate symbol F,
is a from the function space X -> Y, only shows

his arrogance which is in fact ignorance.

Mostowski Collapse schrieb am Freitag, 3. Dezember 2021 um 09:59:04 UTC+1:
> Ask Dan-O-Matik about cardinal arithmetic, he will not have
> any clue. Another notation for the function space A -> B, is B^A.
> And then we have this nice bijection theorem:
>
> P(A) ~ 2^A
>
> Subsequently:
>
> |P(A)| = |2^A|
>
> For example N is countable and 2^N is uncountable. Does Dan-O-Matik
> even have the slightest idea what functions spaces are?
>
> See also:
> Given two sets S and T, the set of all functions from T to S is denoted S^T.
> This fits in with the exponentiation of cardinal numbers, in the sense
> that |S^T| = |S|^|T|, where |X| is the cardinality of X.
> https://en.wikipedia.org/wiki/Exponentiation#Sets_as_exponents
> Mostowski Collapse schrieb am Freitag, 3. Dezember 2021 um 09:40:54 UTC+1:
> > An the winner for going insane is Dan-O-Matik:
> >
> > Dan-O-Matik halucinated his own little crazy world:
> > > Given s0={0}, f:s0-->s0 and ~1=0, we have: f(0)=0 and f(1) is indeterminate (undefined)
> > > Formal Proof: https://www.dcproof.com/FunctionUndefined.htm (23 lines)
> >
> > This is not correct mathematics. If you say f : X -> Y, you
> > want the domain of f to be determined to be X.
> > If you have for f : {0} -> {0}, f(1) inderminate, then dom(f)
> >
> > can be either {0} or {0,1}. You need to have f(1) undefined,
> > so that the domain is always {0}. You can check any standard
> > reference in mathematics, for example:
> >
> > Basic Set Theory - Azriel Levy
> > https://www.amazon.de/dp/0486420795
> >
> > The booklet is only 20 bucks or so.
> > zelos...@gmail.com schrieb am Freitag, 3. Dezember 2021 um 06:24:09 UTC+1:
> > > torsdag 2 december 2021 kl. 15:51:54 UTC+1 skrev timba...@gmail.com:
> > > > On Thursday, December 2, 2021 at 9:18:34 AM UTC-5, Gus Gassmann wrote:
> > > > > On Thursday, 2 December 2021 at 04:50:41 UTC-4, WM wrote:
> > > > > > [...]
> > > > > > Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N.. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
> > > > > It's "Nauchnoe Slovo". Sad to see that you can't even copy the name of the journal properly any longer. Note also that Kolmogorov uses "appeared" (note even "appear", but past tense) in an article on "debates". He also mentions Hilbert's approach, but of course you chose to ignore that entirely. Your selective quotations, taken out of context, are unethical and unprofessional. But we knew that you are a lying, cheating son of a bitch, so, nothing to see here.
> > > > Ummm.... guys, I am in danger of unifying the integer with the continuum. As your numbers turn large do they care about the little end of things? No, they do not. The large values outweight the littler digits. Like siblings the biggest get first dibs, and but for their kindness and respect to the little ones there is not much going for them. Of course we would like an egalitarian number system. But when would you say it is true that one million versus one million and one mark a breaking point? In some base threshing mechanism perhaps, and in that mechanism I might well ask whether two million versus two million and one mark some stark breakpoint? Well then, you've been engaging in radix principles, perhaps.
> > > >
> > > > Interesting possibility: if you did claim to care about sin(x) in N and create some variants on this would ye be recovering a radix system? Why, you'd have your lower digits in no time. Did you want to radix three? Is it any mystery where the next mark is? But for labels these are reusables and they are asking for something that is foreign to your system. Inducing rotational qualities into your values is this easy. Maybe I'll try to expand this out a bit, but clearly we will see that
> > > > sin( 3 ), sin( 4 ), sin( 5 ), sin( 6 ), sin( 7 ), ...
> > > > beg that we put a discrete toothed ratchet in, doesn't it?
> > > > I suppose you could try to go with something else, but it won't be very n-ary. But you see you can't have the exploits of the real values system already. two pi is not a thing here yet, so let's just round down to six and we'll call this beast a dirty re-radixer.
> > > > So you see that when you plug your dirty re-radixer into another dirty re-radixer that you get patterns? Well, maybe I am wrong to be restrictive and insist on an unmixed radix. Mix it up; all the better; shall we let the next one be a radix seven? then the prior was a five? Then this would put the six out at 6!, no? 120... now that is handy. Could you imagine the poor superstitious cultures that would accrue from such a system? Is 360 so far away? Then too, back at two... a sticking point for humans it seems, the first signs of NU and MU as unique appear. And of course you who are N-ary are none the wiser, but for a little dot.
> > > are you going insane?

On Friday, December 3, 2021 at 3:40:54 AM UTC-5, Mostowski Collapse wrote:
> An the winner for going insane is Dan-O-Matik:
>
> Dan-O-Matik halucinated his own little crazy world:
> > Given s0={0}, f:s0-->s0 and ~1=0, we have: f(0)=0 and f(1) is indeterminate (undefined)
> > Formal Proof: https://www.dcproof.com/FunctionUndefined.htm (23 lines)
>
> This is not correct mathematics. If you say f : X -> Y, you
> want the domain of f to be determined to be X.

Here, X would be the domain, and Y the codomain sets for the function f. As it is defined here, f would be undefined for elements outside of the domain X. Really!

> If you have for f : {0} -> {0}, f(1) inderminate, then dom(f)
>
> can be either {0} or {0,1}.

In that case, 1 would be outside of the domain of f. We then say that f(1) is undefined. Deal with it, Jan Burse.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

I guess WM has lost all faith into logic?
https://twitter.com/sarangop/status/1464252740343717897

Isn't it nice to switch on and off actual infinity by a single axiom?

WM schrieb am Donnerstag, 2. Dezember 2021 um 09:50:41 UTC+1:
> There is no possibility of introducing something absolutely uncountable, but by a pure dogma. [T. Skolem: "Über einige Grundlagenfragen der Mathematik", Norske Videnskaps-Akademi, Mat.-naturv. Klasse No. 4, Oslo (1929)]
>
> In order to obtain something absolutely nondenumerable, we would have to have either an absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely nondenumerable number of first-order propositions. [...] It is easy to show that Zermelo's axiom system is not sufficient to provide a complete foundation for the usual theory of sets. [p. 296] I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique." [p. 300f] [T. Skolem: "Some remarks on axiomatized set theory" (1922) quoted in J. van Heijenoort: "From Frege to Gödel – A source book in mathematical logic, 1879-1931", Harvard University Press, Cambridge (1967)]
>
> Of course we know that, e.g., the Axiom of Choice is required for many mathematical theorems (such as: every vector space has a basis), which in turn can be applied in physics. However, on closer inspection it turns out that for all concrete instances that are used, the axiom of choice is not required. The same applies even for the existence of an infinite set: One can use a very constructive, "finitary" form of mathematics that is perfectly sufficient for physics. [...] the old intuition: "if nontrivial set theory, non-constructive mathematics or a non-measurable set is used in an essential way, it cannot be physically relevant". [J. Kellner: "Pitowsky's Kolmogorovian models and super-determinism", arXiv (2016) p. 11f]
>
> Objects whose existence is postulated by this axiom {{Zermelo's axiom of choice}} appeared to be not only useless but sometimes destructive to the simplicity and rigorousness of crucial mathematical theories. [A.N. Kolmogorov: "Modern debates on the nature of mathematics", Nauchae Slovo 6 (1929) pp. 41-54]
>
> Regards, WM