Potential infinity vs actual infinity for the last time

[David Petry]

"Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". -- Poincare

'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)

What Gauss and Poincare are saying is that only the notion of a potential infinity belongs in mathematics. It's really not that hard to understand.

Maybe we should rename potential infinity as "Gauss/Poincare infinity".

[WM]

The finite world-models of present natural science clearly show how the power of the idea of actual infinity has come to an end in classical (modern) physics. In this light the inclusion of the actual infinite into mathematics which explicitly started by the end of the last century with G. Cantor appears disconcerting. In the intellectual overall picture of our century - in particular in view of existentialist philosophy - the actual infinite appears as an anachronism. [...] We introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what way should we ever arrive at infinitely-many countable things? [...] In philosophical terminology we say that the infinite of the number sequence is only potential, i.e., existing only as a possibility. [Lorenzen]

Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by |N. Thus |N is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of "infinite" is "not finished". [Nelson]

A potentially infinite sequence of digits 0.d_1d_2d_3... is never completed and, therefore, cannot define a real number. But that is not required in classical mathematics. There the limit of the sequence is defined as that real number a such that the sequence of rational numbers (a_n) approaches a as closely as desired. For every positive real number eps there exists an index n_eps such that for every index n > n_eps the distance of the term an to the limit a is |a - an| < eps. For this definition it is not necessary (and not possible) that all terms exist. Therefore simple digit structures like 0.111... can be used to denote the limit.

In actual infinity, on the other hand, the limit is the term a following upon all terms a_n with finite indices n. There the complete digit sequence denotes all partial sums. It cannot simultaneously denote the limit a.

Regards, WM

[Dan Christensen]

The notions of actual vs. potential infinity have no place in modern mathematics. They are in the realm of informal philosophical discussions, if that.

In mathematics, just define an infinite set S to be one on which there exists an injective f: S --> S such that f is not surjective. Or something equivalent.

For what I believe to be an innovative development of this notion, starting with the notion of a finite set, see my blog posting, "Infinity: The Story So Far" at http://www.dcproof.wordpress.com


Dan

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

[Dan Christensen]

On Sunday, July 12, 2015 at 5:48:42 AM UTC-4, WM wrote:
> Am Sonntag, 12. Juli 2015 08:44:20 UTC+2 schrieb David Petry:
> > "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". -- Poincare
> >
> > 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)
> >
> > What Gauss and Poincare are saying is that only the notion of a potential infinity belongs in mathematics. It's really not that hard to understand.
> >
> > Maybe we should rename potential infinity as "Gauss/Poincare infinity".
>
> The finite world-models of present natural science clearly show how the power of the idea of actual infinity has come to an end in classical (modern) physics. In this light the inclusion of the actual infinite into mathematics which explicitly started by the end of the last century with G. Cantor appears disconcerting. In the intellectual overall picture of our century - in particular in view of existentialist philosophy - the actual infinite appears as an anachronism. [...] We introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what way should we ever arrive at infinitely-many countable things? [...] In philosophical terminology we say that the infinite of the number sequence is only potential, i.e., existing only as a possibility. [Lorenzen]
>
> Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by |N. Thus |N is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of "infinite" is "not finished". [Nelson]
>
> A potentially infinite sequence of digits 0.d_1d_2d_3... is never completed and, therefore, cannot define a real number.

We are STILL waiting for YOUR OWN definition of what is and is not an infinite set, WM. What seems to be the problem, WM? Are definitions also "rubbish" in your fantasy world?

Dan

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

**********

Really stupid quotes from WM here at sci.math:

"In my system, two different numbers can have the same value."
-- Wolfgang Mueckenheim (WM), sci.math, 2014/10/16

"1+2 and 2+1 are different numbers."
-- Wolfgang Mueckenheim (WM), sci.math, 2014/10/20

"Axioms are rubbish!"
-- Wolfgang Mueckenheim (WM), sci.math, 2014/11/19

[Julio Di Egidio]

On Sunday, July 12, 2015 at 7:44:20 AM UTC+1, David Petry wrote:
> "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". -- Poincare
>
> 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)
>
> What Gauss and Poincare are saying is that only the notion of a potential infinity belongs in mathematics. It's really not that hard to understand.

Infinity enters logic with Zeno and mathematics with Pythagoras already. Indeed, more interesting than what they said would be to know why they said it.

> Maybe we should rename potential infinity as "Gauss/Poincare infinity".

That's a nice idea, but cannot really work: eventually, fool-proof is fool in itself...

Julio

[Julio Di Egidio]

On Sunday, July 12, 2015 at 10:48:42 AM UTC+1, WM wrote:

Just nonsense:

> The finite world-models of present natural science clearly show how the power of the idea of actual infinity has come to an end in classical (modern) physics.

That is pure horseshit... but I won't get into physics.

> In this light the inclusion of the actual infinite into mathematics

In that light, you have no reason to say anything about mathematics.

> which explicitly started by the end of the last century with G. Cantor

Bullshit: Zeno in logic and Pythagoras in mathematics, already. Cantor is set theory, which is far from being the beginning of mathematics.

> in particular in view of existentialist philosophy - the actual infinite appears as an anachronism.

Again complete crap... but I won't get into philosophy.

> [...] We introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what way should we ever arrive at infinitely-many countable things? [...] In philosophical terminology we say that the infinite of the number sequence is only potential, i.e., existing only as a possibility. [Lorenzen]

So Lorenzen agrees that unbounded is not enough.

> Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by |N. Thus |N is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of "infinite" is "not finished". [Nelson]

So Nelson agrees that standard set theory is illogical.

> A potentially infinite sequence of digits 0.d_1d_2d_3... is never completed and, therefore, cannot define a real number.

In finitary mathematics, real numbers are never achieved: only arbitrarily good approximations. But there is not only finitary mathematics, therefore you are simply incompetent and/or unreasonable.

> In actual infinity, on the other hand, the limit is the term a following upon all terms a_n with finite indices n. There the complete digit sequence denotes all partial sums. It cannot simultaneously denote the limit a.

You are as illogical as, but way more sloppy than, standard theory. (EOD.)

Julio

[Virgil]

In article <07949a2a-853c-4d96...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> The finite world-models of present natural science clearly show how the power
> of the idea of actual infinity has come to an end in classical (modern)
> physics.

No one forces physicists to assume any actual infiniteness in their
physics, but also no one allows physicists to dictate what
mathematicians are allowed to assume.
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

[abu.ku...@gmail.com]

my favorite evocation is using base_one:
zero, one, infinity; but,
it is not "hashmarks (or Roman numeralS

[David Petry]

On Sunday, July 12, 2015 at 6:20:07 AM UTC-7, Dan Christensen wrote:


> The notions of actual vs. potential infinity have no place in modern mathematics. They are in the realm of informal philosophical discussions, if that.

English translation: Mathematicians live inside a box. Thinking outside the box is strictly verboten.

[Jim Burns]

On 7/12/2015 2:44 AM, David Petry wrote:

> "Actual infinity does not exist. What we call infinite is only
> the endless possibility of creating new objects no matter
> how many exist already". -- Poincare
>
> 'Infinity is nothing more than a figure of speech which helps us
> talk about limits. The notion of a completed infinity doesn't belong
> in mathematics' -- Gauss (paraphrased)
>
> What Gauss and Poincare are saying is that only the notion of a
> potential infinity belongs in mathematics.

If Poincare is correct that actual infinity does not exist,
then you are finished. Your goal has been accomplished. There is no
actual infinity in mathematics to be removed -- just things
that are called infinite.

Or -- here is another possibility: what you call actual infinity
is not what Poincare calls actual infinity. You apparently can
find actual infinity in use in mathematics, something Poincare
cannot do. (Though your recent dismissal of my arguments because
_the real numbers_ are a potential infinity make me less than
100% certain of that. If the reals are potential, what could be actual?)

If you and Poincare are using the terms potential infinity and
actual infinity differently, don't you think you should at least
make the difference clear? Personally, I would go further, and
not rely on the authority of someone like Poincare to support my
case, if Poincare did not actually support my case (which he
would not if he were talking about something else, and
calling it actual infinity.)

I'm not sure how much of the Gauss paraphrase is Gauss' and
how much is yours. It starts out by saying infinity helps us
talk about limits. It seems to me that Gauss is on my side when
it comes to whether infinity is useful. It could be the paraphrase
comes where you delete how Gauss gets from "Infinity is useful"
to "Actual infinity should not be used". How does he get from
the first to the second?

> It's really not that hard to understand.

I am glad to hear that this is not hard for you to understand.
Please define mathematically potential infinity and actual infinity.

[David Petry]

On Sunday, July 12, 2015 at 4:47:22 PM UTC-7, Jim Burns wrote:

> Please define mathematically potential infinity and actual infinity.

Sure.

When we say that infinity has merely a potential existence, we mean that any mathematical statement involving infinity must be translatable into a statement that does not involve infinity. (in other words, infinity is merely a figure of speech)

Example: the statement sum(k=1..oo) 1/2^k = 1 can be translated into the statement AnEm |sum(k=1..m) 1/2^k - 1| < 1/n, and hence only the notion of potential infinity is used in the first statement.

The claim that there are mathematical statements involving infinity that cannot be translated into statements not involving infinity is equivalent to the claim that infinity has an actual existence.


Just for the record, the example I gave is ever so slightly misleading in a way I don't want to get into now.

[Virgil]

In article <91471b06-0c0e-4163...@googlegroups.com>,

David Petry <david...@gmail.com> wrote:

nity is used in the first statement.
>
> The claim that there are mathematical statements involving infinity that
> cannot be translated into statements not involving infinity is equivalent to
> the claim that infinity has an actual existence.

The definition of a set being finite or being infinite can be and
standardly is expressed in terms not directly involving infinity:
A set S is finite if and only if
every injection from S to S is a surjection.

It is also clear that there are injections from |N to |N that are not
surjections!

[Virgil]

In article <3a9b485b-38e2-4aa4...@googlegroups.com>,

Mathematicians' "boxes" are more properly called axiom systems.
And the games mathematicians play are derived from such axiom systems.

[Jim Burns]

On 7/12/2015 8:37 PM, David Petry wrote:
> On Sunday, July 12, 2015 at 4:47:22 PM UTC-7, Jim Burns wrote:

>> Please define mathematically potential infinity and actual infinity.
>
> Sure.
>
> When we say that infinity has merely a potential existence, we
> mean that any mathematical statement involving infinity must
> be translatable into a statement that does not involve infinity.
> (in other words, infinity is merely a figure of speech)

I admit that this is not what I thought you meant by actual infinity.
I'm not sure, but I suspect that those fine people you have been
quoting, Poincare, Gauss, Feferman, mean something else. Do agree
that they mean something else? If you don't agree, would please address
this, show where they mean the same?

I had thought what you meant by actual infinity would have as an example
an infinite set, whether or not it could be defined without mentioning
infinity, for example, the minimal inductive set N, for which
(1) {} e N
(2) (all x e N)( x U {x} e N )
(3) if A c N, {} e A, and (all x e A)( x U {x} e N ), then A = N

The operations +, * and relation < can all be defined without
mentioning infinity. The integers Z and their operations can be
defined without mentioning infinity. The rationals Q and their
operations can all be defined without mentioning infinity.
The reals R and their operations can all be defined without
mentioning infinity.

In fact, the Axiom of Infinity can be asserted without mentioning
infinity:
(exists A)( {} e A & (all x)( x e A -> x U {x} e A ) )

I have to stop and think of where infinity cannot be translated
into something not referring to infinity.

I suppose the closure of the real line _could_ be such a case,
R U { +infinity, -infinity }, along with the rules extending
+, *, < .

However, I don't have to call the added elements +infinity
and -infinity. I could have called them # and $, and just said
they are these points and follow these rules.
Would R U { #, $ } along with the appropriate rules extending
+ , * , < be potentially infinite?

If it is potential infinity, I don't see what instances of
actual infinity you are complaining about. Could you give examples?

If the infinities are nonetheless there and actual, even though
they're not laeled infinity, how can one tell this mathematically?
How would you pick out # and $ for this purpose, picking out
actual infinity? Do you define infinity for this purpose? How?

Assuming R U { #, $ } is an actual infinity, what makes it
less falsifiable than R (in the sense you use), which apparently
is a potential infinity, since it can be defined and used,
all without mentioning infinity?

[David Petry]

On Sunday, July 12, 2015 at 6:53:23 PM UTC-7, Jim Burns wrote:
> On 7/12/2015 8:37 PM, David Petry wrote:
> > On Sunday, July 12, 2015 at 4:47:22 PM UTC-7, Jim Burns wrote:
>
> >> Please define mathematically potential infinity and actual infinity.
> >
> > Sure.
> >
> > When we say that infinity has merely a potential existence, we
> > mean that any mathematical statement involving infinity must
> > be translatable into a statement that does not involve infinity.
> > (in other words, infinity is merely a figure of speech)
>
> I admit that this is not what I thought you meant by actual infinity.
> I'm not sure, but I suspect that those fine people you have been
> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
> that they mean something else?

The things those guys say about infinity make perfect sense to me, and thus I conclude they must have the same understanding of infinity as I do.

On the other hand, when you start talking about potential infinity or actual infinity, or even falsifiability, almost everything you say strikes me as being very confused gibberish. So I conclude that you and I don't have the same understanding of those terms, and furthermore, I don't accept you as being qualified to even comment on the question of whether I have the same understanding of the issues as the guys you mentioned.

> I had thought what you meant by actual infinity would have as an example
> an infinite set, whether or not it could be defined without mentioning
> infinity,

My first comment would be that the use-mention distinction is really important. Surely you've heard of that. You seem to stumble over that distinction a lot.

Second, I'm getting something of a headache just thinking about the question of how I can communicate with you, when you seem to make a deliberate effort to fail to understand everything I say.

There's a reality out there. It's what we can observe and interact with. We can understand it, and we can have a shared understanding of that reality. Mathematics is a language that can help us understand that reality, and help us communicate with each other based on our shared understanding of that reality. If we don't establish that shared understanding of reality, we will be fighting forever. That's something I don't want.

So as I see it, what you and the Cantorian mathematicians are doing is playing a game in which you can prove your own intellectual superiority, or something like that. And you keep trying to draw me into your game. And I hope you fail at drawing me into your game. I think your game will lead to endless fighting. I also think your game is pulling the whole world down into endless fighting. I'd like to crush your game out of existence.

[Virgil]

In article <aef77c7d-856f-4ffe...@googlegroups.com>,

David Petry <david...@gmail.com> wrote:

> So as I see it, what you and the Cantorian mathematicians are doing is
> playing a game in which you can prove your own intellectual superiority, or
> something like that. And you keep trying to draw me into your game. And I
> hope you fail at drawing me into your game. I think your game will lead to
> endless fighting. I also think your game is pulling the whole world down into
> endless fighting. I'd like to crush your game out of existence.

PURE Mathematics is very like a form of game playing in which one
invents a sets of rules (axiom systems) and sees what can be deduced
from them. Of course pure mathematicians try to create what they view as
"mathematically interesting" axiom systems usually quite regardless of
whether physicists will regard them as physically interesting.

That so many of these mathematical games turn out to to be of such great
interest to physicists and other non-mathematicians is merely a lucky
break!

[WM]

Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:


> I admit that this is not what I thought you meant by actual infinity.
> I'm not sure, but I suspect that those fine people you have been
> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
> that they mean something else? If you don't agree, would please address
> this, show where they mean the same?

Simplest example: Actual infinity is a number aleph_0 that is fixed and larger than every natural number. In the countable cardinal numbers n we have

E aleph_0 A n : n =< aleph_0.

>
> I had thought what you meant by actual infinity would have as an example
> an infinite set, whether or not it could be defined without mentioning
> infinity, for example, the minimal inductive set N, for which
> (1) {} e N
> (2) (all x e N)( x U {x} e N )
> (3) if A c N, {} e A, and (all x e A)( x U {x} e N ), then A = N

This is a potentially infinite set. The interpretation as actually infinite is actually nonsense.

Regards, WM

[Alan Smaill]

WM <wolfgang.m...@hs-augsburg.de> writes:

> Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:
>
>> I admit that this is not what I thought you meant by actual infinity.
>> I'm not sure, but I suspect that those fine people you have been
>> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
>> that they mean something else? If you don't agree, would please address
>> this, show where they mean the same?
>
> Simplest example: Actual infinity is a number aleph_0 that is fixed
> and larger than every natural number.

Simpler example: accepting the set of natural numbers as a completed
entity. That is already an example of actual infinity, isn't it?

WM invariably introduces aleph_0 into the domain of discussion
where it is absent in conventional accounts, eg in the first-order
theory of natural numbers (no sets).

The more to mudddy the waters, no doubt.

>
> Regards, WM

--
Alan Smaill

[Dan Christensen]

On Monday, July 13, 2015 at 4:22:13 AM UTC-4, WM wrote:
> Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:
>
>
> > I admit that this is not what I thought you meant by actual infinity.
> > I'm not sure, but I suspect that those fine people you have been
> > quoting, Poincare, Gauss, Feferman, mean something else. Do agree
> > that they mean something else? If you don't agree, would please address
> > this, show where they mean the same?
>
> Simplest example: Actual infinity is a number aleph_0 that is fixed and larger than every natural number. In the countable cardinal numbers n we have
>

The set of all natural numbers is the simplest example of an infinite set. While philosophers may make use of informal notions of actual vs. potential infinity for whatever reasons, they are not useful distinctions in modern mathematics. Useful distinctions in modern mathematics include countable and uncountable infinities, e.g. the set of natural numbers N is countably infinite, while the power set of N is uncountably infinite.

> > I had thought what you meant by actual infinity would have as an example
> > an infinite set, whether or not it could be defined without mentioning
> > infinity, for example, the minimal inductive set N, for which
> > (1) {} e N
> > (2) (all x e N)( x U {x} e N )
> > (3) if A c N, {} e A, and (all x e A)( x U {x} e N ), then A = N
>
> This is a potentially infinite set. The interpretation as actually infinite is actually nonsense.
>

We are STILL waiting for your formal definition of ANY kind of infinity (potential, actual or otherwise). What seems to be the problem, WM?

Dan

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

**********

Really stupid quotes from WM here at sci.math:

"In my system, two different numbers can have the same value."
-- Wolfgang Mueckenheim (WM), sci.math, 2014/10/16

"1+2 and 2+1 are different numbers."
-- Wolfgang Mueckenheim (WM), sci.math, 2014/10/20

"Axioms are rubbish!"
-- Wolfgang Mueckenheim (WM), sci.math, 2014/11/19

A word of caution to students: Do not attempt to use WM's "system" in any course work in any high school, college or university on the planet. You will fail miserably.

[Virgil]

In article <2160d410-a74f-404b...@googlegroups.com>,

In proper mathematics, a set S is, by definition, finite if and only if
every injection from S to S is a surjection and is infinite otherwise,
and there is no definition of a set being potentially-infinite.

And until WM can provide us with a proper definition of a set being
potentially infinite while not infinite in the above sense, there can be
no such separate thing as potential infiniteness.

[Jim Burns]

On 7/12/2015 11:49 PM, David Petry wrote:
> On Sunday, July 12, 2015 at 6:53:23 PM UTC-7, Jim Burns wrote:

>> I had thought what you meant by actual infinity would have as an
>> example an infinite set, whether or not it could be defined without
>> mentioning infinity,
>
> My first comment would be that the use-mention distinction is really
> important. Surely you've heard of that. You seem to stumble over
> that distinction a lot.

It would be helpful if you explained your comment. Where does
the use-mention distinction come into play here? You seem to be
making a criticism here, but that's all I can tell, and that only
because of your tone.

Did you notice that _right here_ I am not saying what you believe,
I am saying what I _used to think_ you believed _in the past_ .
You know, like _not now_ .

I am applying your definition,

> When we say that infinity has merely a potential existence, we
> mean that any mathematical statement involving infinity must
> be translatable into a statement that does not involve infinity.
> (in other words, infinity is merely a figure of speech)

It looks to me as though most or all of mathematical infinity
is potential infinity, under your definition.

The text seems clear enough to me. Also, my interpretation
matches the example you gave. And it also explains your surprising
declaration that my argument about the real numbers is really
about potential infinity.

For example, you give an example of an infinite sum, and show that
it can be written without reference to infinity. You say this
means that infinity is only a potential infinity.

So, I presented you with another example, a definition of the minimal
inductive set. This too can be written without reference to infinity.
Is the minimal inductive set a potential infinity? Please just
tell me if it is or isn't. If you wish to, go ahead and add an
unexplained reference to something like the use-mention distinction,
but please remember to tell me yes or no.

For clarity:

The set A is inductive iff
{} e A & (all x)( x e A -> (x U {x}) e A )

It is a theorem that every inductive set A contains an inductive
subset which is a subset of every inductive subset of A.
Call this the minimal inductive subset of A

Two inductive sets A and B have the same set as their respective
minimal indictive subsets. Call this unique minimal inductive
subset _the_ minimal inductive set N.

The set N together with the appropriately defined constant {}.
operations *, + and relation < is a model of the Peano axioms.

>
> Second, I'm getting something of a headache just thinking about the
> question of how I can communicate with you, when you seem to make a
> deliberate effort to fail to understand everything I say.

With regard to communicating with me: How about answering my
questions?

[David Petry]

On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:
> On 7/12/2015 11:49 PM, David Petry wrote:


> > When we say that infinity has merely a potential existence, we
> > mean that any mathematical statement involving infinity must
> > be translatable into a statement that does not involve infinity.
> > (in other words, infinity is merely a figure of speech)
>
> It looks to me as though most or all of mathematical infinity
> is potential infinity, under your definition.

The constructivists have done a pretty good job of showing all, or virtually all, of the mathematics that helps us reason about the real world (e.g. physics) can be done constructively (i.e. using only the notion of a potential infinity). It's very debateable whether I should have written "or virtually all".

[David Petry]

On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:


> Is the minimal inductive set a potential infinity? Please just
> tell me if it is or isn't.

I don't think you can prove that it exists without the axiom of infinity.

Did you read the quote from Ed Nelson that WM gave us?

**start quote**

Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by |N. Thus |N is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of "infinite" is "not finished".

**end quote**

Ed Nelson is mathematics professor at Princeton, and he is well regarded even if he may be a little eccentric. He wrote a lot about the notions of potential and actual infinity.

http://kolany.pl/KNMaT/r%C3%B3%C5%BCno%C5%9Bci/PA/dane/2.%20elem.pdf

[Ross A. Finlayson]

Of a mathematical physics,
with physical objects as
mathematical objects, the
functions among them are
also physical objects, and
there are infinitely many
of them. (The Universe is
infinite.)

The universe of mathematical
objects would be its own
powerset.

There are a variety of known
extra-classical effects in the
micro and macro (and not the
finite).

Transfinite cardinals contribute
no analytical character to the
reals, but infinitesimals do, and
these are also fundamental features
of parastatistics.


Yer finite math is weak.

[Virgil]

In article <fdfb2f42-42f2-4bc0...@googlegroups.com>,

David Petry <david...@gmail.com> wrote:

> On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:
> > On 7/12/2015 11:49 PM, David Petry wrote:

>
> The constructivists have done a pretty good job of showing all, or virtually
> all, of the mathematics that helps us reason about the real world (e.g.

> physics) can be done constructively.

And absoutely all of it can be done and has been done more axiomatically
based on some form of an axiom of infinity!

But physics hardly uses all of mathematics. Most of number theory, based
on an actually infinite set of naturals, including the encoding methods
essential to conducting all modern businesses on line, is not now and
never has been used by physicists to do their physics.

So if we were to remove from modern mathematics only those parts of
mathematics NOT specifically used by physics, the world economy would
totally crash.

Thus the hypothesis that mathematics needs physics is falsified.

[Ross A. Finlayson]

Information theory is a quite
regular part of physics.

Perhaps you've heard of entropy.

[Virgil]

In article <1f7a5309-3ee8-45d2...@googlegroups.com>,

David Petry <david...@gmail.com> wrote:

> On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:
>
>
> > Is the minimal inductive set a potential infinity? Please just
> > tell me if it is or isn't.
>
> I don't think you can prove that it exists without the axiom of infinity.

Equally you can't prove it doesn't witout some axion which is no part of
any standard axiomatization of sets.

>
> Did you read the quote from Ed Nelson that WM gave us?
>

> Numerals constitute a potential infinity.

Not unless one puts a finite cap on the number of digits allowed in one.

If one allows always another digit to be appended to any numeral then
set of numerals is actually infinite.

[John Dawkins]

In article <1f7a5309-3ee8-45d2...@googlegroups.com>,
David Petry <david...@gmail.com> wrote:

Nelson was the author of perhaps the shortest paper ever published "A
Proof of Liouville's Theorem", comprising one paragraph in the
Proceedings of The AMS in 1961.

He died last September:

<http://www.princeton.edu/main/news/archive/S41/11/36I14/index.xml?sectio
n=topstories>

--
J.

[WM]

Am Montag, 13. Juli 2015 14:40:06 UTC+2 schrieb Alan Smaill:
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
> > Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:
> >
> >> I admit that this is not what I thought you meant by actual infinity.
> >> I'm not sure, but I suspect that those fine people you have been
> >> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
> >> that they mean something else? If you don't agree, would please address
> >> this, show where they mean the same?
> >
> > Simplest example: Actual infinity is a number aleph_0 that is fixed
> > and larger than every natural number.
>
> Simpler example: accepting the set of natural numbers as a completed
> entity. That is already an example of actual infinity, isn't it?

Only calling it "actual" is useless. Behind the word there must be fact. The simplest fact is "aleph_0 elements".

>
> WM invariably introduces aleph_0 into the domain of discussion
> where it is absent in conventional accounts, eg in the first-order
> theory of natural numbers (no sets).

No actual infinity there.

Regards, WM

[WM]

Am Montag, 13. Juli 2015 18:31:38 UTC+2 schrieb Virgil:


> And until WM can provide us with a proper definition of a set being
> potentially infinite while not infinite in the above sense, there can be
> no such separate thing as potential infiniteness.

Potentially infinite is the inductive set (defined by Peano, Zermelo, or v. Neumann) without a measure aleph_0 that is a fixed quantity and gives the number of elements of the set.

Potentially infinite is a digit sequence that does not define a real number.

Regards, WM

[Virgil]

In article <792d9460-dfb8-4d8a...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Montag, 13. Juli 2015 14:40:06 UTC+2 schrieb Alan Smaill:
> > WM <wolfgang.m...@hs-augsburg.de> writes:
> >
> > > Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:
> > >
> > >> I admit that this is not what I thought you meant by actual infinity.
> > >> I'm not sure, but I suspect that those fine people you have been
> > >> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
> > >> that they mean something else? If you don't agree, would please address
> > >> this, show where they mean the same?
> > >
> > > Simplest example: Actual infinity is a number aleph_0 that is fixed
> > > and larger than every natural number.
> >
> > Simpler example: accepting the set of natural numbers as a completed
> > entity. That is already an example of actual infinity, isn't it?
>
> Only calling it "actual" is useless.

That it points out the falseness of WM's arguments makes it useful!

> > WM invariably introduces aleph_0 into the domain of discussion
> > where it is absent in conventional accounts, eg in the first-order
> > theory of natural numbers (no sets).
>
> No actual infinity there.

Does WM still claim that the number of natural numbers is a finite
number?

Does WM still claims to have a set theory in which there is no set of
all natural numbers? Until he can produce an axiom system for it, he is,
as usual, merely blowing hot air!

[Virgil]

In article <10657a59-47e3-4210...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Montag, 13. Juli 2015 18:31:38 UTC+2 schrieb Virgil:
>
>
> > And until WM can provide us with a proper definition of a set being
> > potentially infinite while not infinite in the above sense, there can be
> > no such separate thing as potential infiniteness.
>
> Potentially infinite is the inductive set (defined by Peano, Zermelo, or v.
> Neumann)

Then actually infinite sets inductive sets like Peano sets are also
potentially infinite?

Note that the very definition of a set S being finite is that every
injection from S to S must be a surjection. Since this is false for
every Peano set, no Peano set can be counted as finite

> Potentially infinite is a digit sequence that does not define a real number.

Outside of WM's worthless world of WMytheology, a mere digit sequence
can only define a natural number. One needs at least one radix point or
one negative sign to name any other numbers!

[abu.ku...@gmail.com]

joke in base_one (not hasmarks [Roman numeralS:
zero, one, infifity.

[WM]

Am Dienstag, 14. Juli 2015 15:58:21 UTC+2 schrieb Virgil:

> > Potentially infinite is the inductive set (defined by Peano, Zermelo, or v.
> > Neumann)
>
> Then actually infinite sets inductive sets like Peano sets are also
> potentially infinite?

They are actually not actually infinite. That is only the confusion of matheologians.

>
> > Potentially infinite is a digit sequence that does not define a real number.
>

> a mere digit sequence
> can only define a natural number. One needs at least one radix point or
> one negative sign to name any other numbers!

For an irrational number one needs more than available. But that is easily proven by the fact that hitherto everybody failed and in future this situation will persist. (But counterfactual claim in absurdum will persist too.)

Regards, WM

[Virgil]

In article <552d3167-8be1-4869...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Dienstag, 14. Juli 2015 15:58:21 UTC+2 schrieb Virgil:
>
> > > Potentially infinite is the inductive set (defined by Peano, Zermelo, or
> > > v.
> > > Neumann)
> >
> > Then actually infinite sets inductive sets like Peano sets are also
> > potentially infinite?
>
> They are actually not actually infinite.

Do they allow self-injections that are not self-surjections?
Only sets like Peano sets which do are not finite in any proper set
theory!

> >
> > > Potentially infinite is a digit sequence that does not define a real
> > > number.
> >
> > a mere digit sequence
> > can only define a natural number. One needs at least one radix point or
> > one negative sign to name any other numbers!
>
> For an irrational number one needs more than available.

In binary, one only needs two different digits for any real number!

[Port563]

"WM" <wolfgang.m...@hs-augsburg.de> wrote in message
news:552d3167-8be1-4869...@googlegroups.com...

> For an irrational number one needs more than available. But that is easily
> proven by the fact that hitherto everybody failed and in future this
> situation will persist. (But counterfactual claim in absurdum will persist
> too.)

Is the above scribble produced by a sentient human, or by an automated
gobbledygook generator a la Plutonium?

[David Petry]

I believe in giving people the benefit of the doubt. It is possible that the "scribble" started off as a coherent statement in German, and then was run through Google Translate.

Maybe I'm being silly.

[WM]

It has been produced by an envious anonymous coward, who snipped the context including the subject of the sentence:

For wrinting an irrational number one needs more digits than are available. This situation will never change.

Regards, WM

[Jim Burns]

On 7/14/2015 12:29 AM, David Petry wrote:
> On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:

>> Is the minimal inductive set a potential infinity? Please just
>> tell me if it is or isn't.
>
> I don't think you can prove that it exists without the axiom
> of infinity.

_If_ the minimal inductive set existed, _would_ it be a
potential infinity? This seems like a very straightforward,
relevant question, with a very easy, short answer, for someone
(like you) who finds all this so very easy.

I'm not sure your point about the axiom of infinity matters,
though. Using your definition[1], the axiom of infinity appears
to me to be referring to potential infinity. This looks to me
like a statement that does not involve infinity:
(exists A)( {} e A & (all x)( x e A -> (x U {x}) e A ) )

Perhaps you need to clarify your definition, or maybe
alter it to reflect what you really mean, or maybe the
axiom of infinity does really refer to _your_ potential infinity.

I suspect this last option is correct, no matter how surprising
it may sound to you. You wrote a couple paragraphs a few
posts back about your motives, and seems that it is the undefined
nature of infinity that gets you going. But infinity is not
undefined, not in ZFC at least, which I think is the gold standard
for mathematical-foundation-talk currently. Sets are undefined,
is-an-element-of is undefined, and I think that's it. Everything
else is defined from those primitive concepts. Infinity
is not primitive in ZFC.

>
> Did you read the quote from Ed Nelson that WM gave us?

No, I don't read much WM.

Ed Nelson's idea of what potential infinity is looks to me
to be different from yours, not just phrased differently,
but not equivalent. You talk about translatability,
Ed Nelson talks about sets like N containing all of
a "process" that has no end point. These might be
equivalent, in the sense of identifying the same statements
or objects as using actual infinity, but I have mentioned
that N (represented by the minimal inductive set)
seems to be one of _your_ potential infinities, and
Ed Nelson gives N as an example of one of _his_ actual
infinities.

>
> **start quote**
> Numerals constitute a potential infinity. Given any numeral,
> we can construct a new numeral by prefixing it with S. Now
> imagine this potential infinity to be completed. Imagine the
> inexhaustible process of constructing numerals somehow to have
> been finished, and call the result the set of all numbers,
> denoted by |N. Thus |N is thought to be an actual infinity
> or a completed infinity. This is curious terminology, since
> the etymology of "infinite" is "not finished".
> **end quote**
>
> Ed Nelson is mathematics professor at Princeton, and he is well
> regarded even if he may be a little eccentric. He wrote a lot
> about the notions of potential and actual infinity.
>
> http://kolany.pl/KNMaT/r%C3%B3%C5%BCno%C5%9Bci/PA/dane/2.%20elem.pdf
>

[1]<q>

> When we say that infinity has merely a potential existence, we
> mean that any mathematical statement involving infinity must
> be translatable into a statement that does not involve infinity.
> (in other words, infinity is merely a figure of speech)

</q>
<91471b06-0c0e-4163...@googlegroups.com>

[WM]

Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:


> > > a mere digit sequence
> > > can only define a natural number. One needs at least one radix point or
> > > one negative sign to name any other numbers!
> >
> > For an irrational number one needs more than available.
>
> In binary, one only needs two different digits for any real number!

Then define an irrational number of your choice by digits. Remember: "LIM" and "SUM" and "..." are not digits.

Regards, WM

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 11:57:23 AM UTC+1, Jim Burns wrote:
> On 7/14/2015 12:29 AM, David Petry wrote:
> > On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:

<snip

> >> Is the minimal inductive set a potential infinity? Please just
> >> tell me if it is or isn't.
> >
> > I don't think you can prove that it exists without the axiom
> > of infinity.
>
> _If_ the minimal inductive set existed, _would_ it be a
> potential infinity? This seems like a very straightforward,
> relevant question, with a very easy, short answer, for someone
> (like you) who finds all this so very easy.
>
> I'm not sure your point about the axiom of infinity matters,
> though. Using your definition[1], the axiom of infinity appears
> to me to be referring to potential infinity. This looks to me
> like a statement that does not involve infinity:
> (exists A)( {} e A & (all x)( x e A -> (x U {x}) e A ) )

The confusion only exists thanks to standard theory: of course an inductive sequence is potentially infinite and necessarily an infinite set is actually infinite. But a set that pops into existence by collecting a potential infinite sequence is first of all invalid nonsense, then broken mathematics.

Julio

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 12:10:30 PM UTC+1, WM wrote:
> Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:

<snip>

> > In binary, one only needs two different digits for any real number!
>
> Then define an irrational number of your choice by digits.

Liars, morons, spammers...

Julio

[Virgil]

In article <797b5a96-4886-4b78...@googlegroups.com>,

One can certainly define an actually infinite SEQUENCE of decimal digits
with decimal point by using "Sum"!

E.g., "SUM_(n in |N) 1/10^(n!) in base ten" does so! And the result
cannot also be expressed as a ratio of integers.

[Virgil]

On Wednesday, July 15, 2015 at 12:10:30 PM UTC+1, WM wrote:
> Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:
<snip>
> > In binary, one only needs two different digits for any real number!
>
> Then define an irrational number of your choice by digits.

A rqdix point followed by a sequence of binary digits which are 1 at and
only at an n!-th position following that radix point.

[Virgil]

In article <fe9e4ec2-87ca-4e66...@googlegroups.com>,

In any base, one only needs two digits, 0 and 1:

In any base, b, Sum_(n in |N) 1/b^(n!) is irrational.

[Virgil]

In article <55A63C90...@osu.edu>, Jim Burns <burn...@osu.edu>
wrote:

> On 7/14/2015 12:29 AM, David Petry wrote:
> > On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:
>
> >> Is the minimal inductive set a potential infinity? Please just
> >> tell me if it is or isn't.
> >
> > I don't think you can prove that it exists without the axiom
> > of infinity.
>
> _If_ the minimal inductive set existed, _would_ it be a
> potential infinity? This seems like a very straightforward,
> relevant question, with a very easy, short answer, for someone
> (like you) who finds all this so very easy.
>
> I'm not sure your point about the axiom of infinity matters,
> though. Using your definition[1], the axiom of infinity appears
> to me to be referring to potential infinity. This looks to me
> like a statement that does not involve infinity:
> (exists A)( {} e A & (all x)( x e A -> (x U {x}) e A ) )

A set A is finite by definition if every injection from A to A is
necessarily also a surjection from A ONto A.

If (exists A)({} e A & (all x)(x e A -> (x U {x}) e A ))
then x -> (x U {x}) is an injection from that A to A which is
NOT a surjection, so A is NOT finite!

[Virgil]

In article <aa21dc63-5a88-4397...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:


> The confusion only exists thanks to standard theory: of course an inductive
> sequence is potentially infinite

Any inductive sequence whose terms are regarded as forming a set must
form an actually non-finite set. The standard definition of a set, S,

being finite is that every injection from S to S must be a surjection.

Since the terms of an inductive sequence regarded as members of a set
lack this property, such a set is non-finite, which in ordinary
mathematics is the same as being infinite.

[Alan Smaill]

David Petry <david...@gmail.com> writes:

> On Monday, July 13, 2015 at 5:31:26 PM UTC-7, Jim Burns wrote:

>> On 7/12/2015 11:49 PM, David Petry wrote:
>
>

>> > When we say that infinity has merely a potential existence, we
>> > mean that any mathematical statement involving infinity must
>> > be translatable into a statement that does not involve infinity.
>> > (in other words, infinity is merely a figure of speech)
>>

>> It looks to me as though most or all of mathematical infinity
>> is potential infinity, under your definition.

>
> The constructivists have done a pretty good job of showing all, or
> virtually all, of the mathematics that helps us reason about the real

> world (e.g. physics) can be done constructively (i.e. using only the
> notion of a potential infinity). It's very debateable whether I
> should have written "or virtually all".

Which constructivists are you thinking about?

The intuitionists for example are happy to consider the formation of the
collection of natural numbers as a basic operation (without getting into
higher cardinalities). Feferman likewise takes the collection of
naturals as basic, but not arbitrary subsets (and he describes this work
as explicit maths rather than constructive).


--
Alan Smaill

[Alan Smaill]

WM <wolfgang.m...@hs-augsburg.de> writes:

> Am Montag, 13. Juli 2015 14:40:06 UTC+2 schrieb Alan Smaill:
>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>
>> > Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:
>> >
>> >> I admit that this is not what I thought you meant by actual infinity.
>> >> I'm not sure, but I suspect that those fine people you have been
>> >> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
>> >> that they mean something else? If you don't agree, would please address
>> >> this, show where they mean the same?
>> >
>> > Simplest example: Actual infinity is a number aleph_0 that is fixed
>> > and larger than every natural number.
>>
>> Simpler example: accepting the set of natural numbers as a completed
>> entity. That is already an example of actual infinity, isn't it?
>
> Only calling it "actual" is useless.

Just answer the question, please.

Accepting the set of natural numbers as a completed
entity, is already an example of actual infinity, isn't it?


>
> Regards, WM

--
Alan Smaill

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 4:33:27 PM UTC+1, Virgil wrote:
> In article <aa21dc63-5a88-4397...@googlegroups.com>,

> Julio Di Egidio <j***@diegidio.name> wrote:
>
> > The confusion only exists thanks to standard theory: of course an inductive
> > sequence is potentially infinite
>
> Any inductive sequence whose terms are regarded as forming a set must
> form an actually non-finite set.

For the umpteenth time: An standard sequence, inductive or otherwise, just cannot form a set. You need to take the limit of that sequence, then the set contains the limit as well.

Julio

[YBM]

An umpteenth mistake is still a mistake, crank.

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 4:35:07 PM UTC+1, Alan Smaill wrote:

> Which constructivists are you thinking about?
>
> The intuitionists for example

The matter here is not directly related to different logics: if we are, we are rather concerned with "constructive vs. axiomatic" approaches.

Of course, the term "constructivism" and related has been so badly abused that I'd suggest nobody uses it anymore, unless they first give a definition...

Julio

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 4:40:07 PM UTC+1, Alan Smaill wrote:

> Just answer the question, please.
>
> Accepting the set of natural numbers as a completed
> entity, is already an example of actual infinity, isn't it?

The question is misguided: the point is that there cannot be a set of all and only the natural numbers.

Julio

[Virgil]

WM <wolfgang.m...@hs-augsburg.de> writes:

> Am Montag, 13. Juli 2015 14:40:06 UTC+2 schrieb Alan Smaill:
>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>
>> > Am Montag, 13. Juli 2015 03:53:23 UTC+2 schrieb Jim Burns:
>> >
>> >> I admit that this is not what I thought you meant by actual
infinity.
>> >> I'm not sure, but I suspect that those fine people you have been
>> >> quoting, Poincare, Gauss, Feferman, mean something else. Do agree
>> >> that they mean something else? If you don't agree, would please
address
>> >> this, show where they mean the same?
>> >
>> > Simplest example: Actual infinity is a number aleph_0 that is fixed
>> > and larger than every natural number.
>>
>> Simpler example: accepting the set of natural numbers as a completed
>> entity. That is already an example of actual infinity, isn't it?
>
> Only calling it "actual" is useless.

Calling it anything else is even more useless.

A set is by definition actually finite if and only if every injection
from it to itself is a surjection.

Or does WM claim to have a better definition of set finiteness?

If so WM has certainly never produced it here!

Thus the negation of a set being finite must be that the set has at
least on injection to itself that is not a surjection, and a standard
prefix meaning "not" is "in", thus "infinite" of a set means not finite,
or requiring an injection of the set to itself that is NOT a surjection.

For the set of naturals, |N, the mapping n -> n+1 is an injection that
is not a surjection, so that |N by that definition is infinite.

Thus "actual" infiniteness has a proper unambiguous definition,
but mere "potential" infiniteness doesn't!

[Julio Di Egidio]

But you cannot say where the mistake is, idiot.

Julio

[Virgil]

In article <9a7d3850-cf64-4b6f...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

Whyever not? Is there no way of telling of an arbitrary object whether
it is or is not a natural number?

Unless there are such objects whose naturalness/unnaturalness is forever
unknowable, such a set must exist, at least in any proper set theory.

[Virgil]

In article <34d879a6-09a4-430e...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

You many need to, I certainly do not!

Having a set in my world merely requires being able to distinguish its
members from non-members, so unless the limit of a sequence is also a
term of that sequence, I can have all members of such a sequence in a
set which excludes any and all limits to it.

In my world, {1/n: n in |N} is a set/sequence which does not contain 0.

[Virgil]

In article <cafc8a87-7c10-4b4f...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

Your mother made it!


Given any set S and any member s of that set, then S without s is also a
set, so if a sequence plus its limit can form a set, so can the sequence
witout it limit form a set.

At least outside of Julio's perversion of WM's worthless world of
WMytheology.

[Alan Smaill]

I agree that there is an important difference between a generating
process and any potential or actual collection of generated elements.

But what do you mean by the "limit of the sequence", and why is it
needed?

It looks like you think that there is no way to consider
{1, 2, 3, ...} (without a last element) as a set, but
{1, 2, 3, ... , w} would be OK. Or maybe not.
Can you clarify?


>
> Julio

--
Alan Smaill

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Wednesday, July 15, 2015 at 4:35:07 PM UTC+1, Alan Smaill wrote:
>
>> Which constructivists are you thinking about?
>>
>> The intuitionists for example
>
> The matter here is not directly related to different logics: if we
> are, we are rather concerned with "constructive vs. axiomatic"
> approaches.

There are several axiomatic accounts of constructive mathematics
(Martin-Lof for example). In these cases, the philosophical
analysis led faily quickly to alternative formalisms.

> Of course, the term "constructivism" and related has been so badly
> abused that I'd suggest nobody uses it anymore, unless they first give
> a definition...
>
> Julio

--
Alan Smaill

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 5:35:07 PM UTC+1, Alan Smaill wrote:

> Julio Di Egidio <j***@diegidio.name> writes:
> > On Wednesday, July 15, 2015 at 4:35:07 PM UTC+1, Alan Smaill wrote:
> >
> >> Which constructivists are you thinking about?
> >>
> >> The intuitionists for example
> >
> > The matter here is not directly related to different logics: if we
> > are, we are rather concerned with "constructive vs. axiomatic"
> > approaches.
>
> There are several axiomatic accounts of constructive mathematics
> (Martin-Lof for example).

Exactly: which is why I was concurring with the question but objecting to the example.

> In these cases, the philosophical
> analysis led faily quickly to alternative formalisms.

The philosophical analyses are for the most part not even wrong, certainly beside the point: how to get from finite to infinite, if that can be done at all.

[Julio Di Egidio]

On Wednesday, July 15, 2015 at 5:35:07 PM UTC+1, Alan Smaill wrote:

> Julio Di Egidio <j***@diegidio.name> writes:
> > On Wednesday, July 15, 2015 at 4:33:27 PM UTC+1, Virgil wrote:
> >> In article <aa21dc63-5a88-4397...@googlegroups.com>,
> >> Julio Di Egidio <j***@diegidio.name> wrote:
> >>
> >> > The confusion only exists thanks to standard theory: of course an
> >> > inductive sequence is potentially infinite
> >>
> >> Any inductive sequence whose terms are regarded as forming a set
> >> must form an actually non-finite set.
> >
> > For the umpteenth time: An standard sequence, inductive or otherwise,
> > just cannot form a set. You need to take the limit of that sequence,
> > then the set contains the limit as well.
>
> I agree that there is an important difference between a generating
> process and any potential or actual collection of generated elements.

That is patent nonsense, thank you. For the umpteenth time: a never finished gathering process would be a potentially infinite collection, and that is different from an actually infinite collection, i.e. the putatively completed process. In particular, a standard inductive sequence is a potential infinity, an infinite set is an actual infinity.

> But what do you mean by the "limit of the sequence", and why is it
> needed?

Because that is all we have, structural limits. Just standard theory gets there by magic.

> It looks like you think that there is no way to consider
> {1, 2, 3, ...} (without a last element) as a set, but
> {1, 2, 3, ... , w} would be OK. Or maybe not.
> Can you clarify?

I did not say there is no way, what I said is that standard theory, Peano already, is illogical then broken.

Julio

[Robin Chapman]

One "mistake" is the use of the phrase "the limit" -- of course
not all sequences have limits. But surely this isn't
a mistake but merely a clumsy attempt at legerdemain.

[Julio Di Egidio]

"Robin Chapman" <R.J.C...@exeter.ac.uk> wrote in message
news:mo642s$vnb$1...@dont-email.me...

Still more fallacies... Keep going by yourself, I will be busy vomiting for
a while.

Julio

[Virgil]

In article <mo660f$7tm$1...@dont-email.me>,

"Julio Di Egidio" <ju...@diegidio.name> wrote:

> I will be busy vomiting for a while.

This time, please don't post it!

[Virgil]

In article <2bccfdf5-b268-437c...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Wednesday, July 15, 2015 at 5:35:07 PM UTC+1, Alan Smaill wrote:
> > Julio Di Egidio <j***@diegidio.name> writes:
> > > On Wednesday, July 15, 2015 at 4:33:27 PM UTC+1, Virgil wrote:
> > >> In article <aa21dc63-5a88-4397...@googlegroups.com>,
> > >> Julio Di Egidio <j***@diegidio.name> wrote:
> > >>
> > >> > The confusion only exists thanks to standard theory: of course an
> > >> > inductive sequence is potentially infinite
> > >>
> > >> Any inductive sequence whose terms are regarded as forming a set
> > >> must form an actually non-finite set.
> > >
> > > For the umpteenth time: An standard sequence, inductive or otherwise,
> > > just cannot form a set. You need to take the limit of that sequence,
> > > then the set contains the limit as well.
> >
> > I agree that there is an important difference between a generating
> > process and any potential or actual collection of generated elements.
>
> That is patent nonsense, thank you. For the umpteenth time: a never finished
> gathering process would be a potentially infinite collection

The "GATHERING PROCESS" of a function with domain |N is "finished" as
soon as it is defined! As is any fuction with a Peano set as its domain!

>
> > But what do you mean by the "limit of the sequence", and why is it
> > needed?
>
> Because that is all we have

What about sequences without limits? Many define sequences don't have
limits but are still defined sequences! And we can certainly define a
sequence without ever knowing whether it will have a limit or not!

>
> > It looks like you think that there is no way to consider
> > {1, 2, 3, ...} (without a last element) as a set, but
> > {1, 2, 3, ... , w} would be OK. Or maybe not.
> > Can you clarify?
>
> I did not say there is no way, what I said is that standard theory, Peano
> already, is illogical then broken.

That you say things are false in no way proves them false, and absent
your irrelevant assumptions that standard mathematics is false, nothing
makes it false!

[Virgil]

In article <d8094824-da25-433c...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Wednesday, July 15, 2015 at 5:35:07 PM UTC+1, Alan Smaill wrote:
> > Julio Di Egidio <j***@diegidio.name> writes:
> > > On Wednesday, July 15, 2015 at 4:35:07 PM UTC+1, Alan Smaill wrote:
> > >
> > >> Which constructivists are you thinking about?
> > >>
> > >> The intuitionists for example
> > >
> > > The matter here is not directly related to different logics: if we
> > > are, we are rather concerned with "constructive vs. axiomatic"
> > > approaches.
> >
> > There are several axiomatic accounts of constructive mathematics
> > (Martin-Lof for example).
>
> Exactly: which is why I was concurring with the question but objecting to the
> example.
>
> > In these cases, the philosophical
> > analysis led faily quickly to alternative formalisms.
>
> The philosophical analyses are for the most part not even wrong,

Certainly yours aren't!

[abu.ku...@gmail.com]

correction;
zero <comma one <comma in-finity

[WM]

Am Mittwoch, 15. Juli 2015 18:06:28 UTC+2 schrieb Virgil:
> In article <9a7d3850-cf64-4b6f...@googlegroups.com>,
> Julio Di Egidio <ju...@diegidio.name> wrote:


> > The question is misguided: the point is that there cannot be a set of all and
> > only the natural numbers.
>
> Whyever not? Is there no way of telling of an arbitrary object whether
> it is or is not a natural number?

But every arbitrary object that comes to your face belongs to a finite set.

Regards, WM

[WM]

Am Mittwoch, 15. Juli 2015 17:40:07 UTC+2 schrieb Alan Smaill:


> Accepting the set of natural numbers as a completed
> entity, is already an example of actual infinity, isn't it?
>

It is paying a lip service.

Regards, WM

[WM]

Am Mittwoch, 15. Juli 2015 16:54:55 UTC+2 schrieb Virgil:


> > For wrinting an irrational number one needs more digits than are available.
> > This situation will never change.
>
> In any base, one only needs two digits, 0 and 1:
>
> In any base, b, Sum_(n in |N) 1/b^(n!) is irrational.

That it is not a digit sequence but a finite formula.

Regards, WM

[WM]

Am Mittwoch, 15. Juli 2015 16:49:27 UTC+2 schrieb Virgil:
> On Wednesday, July 15, 2015 at 12:10:30 PM UTC+1, WM wrote:
> > Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:
> <snip>
> > > In binary, one only needs two different digits for any real number!
> >
> > Then define an irrational number of your choice by digits.
>
> A rqdix point followed by a sequence of binary digits which are 1 at and
> only at an n!-th position following that radix point.

n! is not a digit. Write digits, only digits, and nothing but digits! Fail to produce an irrational number.

Regards, WM

[WM]

Am Mittwoch, 15. Juli 2015 16:40:17 UTC+2 schrieb Virgil:
> In article <797b5a96-4886-4b78...@googlegroups.com>,

> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
> > Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:
> >
> >

> > > > > a mere digit sequence
> > > > > can only define a natural number. One needs at least one radix point or
> > > > > one negative sign to name any other numbers!
> > > >
> > > > For an irrational number one needs more than available.

> > >
> > > In binary, one only needs two different digits for any real number!
> >

> > Then define an irrational number of your choice by digits. Remember: "LIM"
> > and "SUM" and "..." are not digits.
>
> One can certainly define an actually infinite SEQUENCE of decimal digits
> with decimal point by using "Sum"!

But one cannot define an irrational number by merely listing digits.

Regards, WM

[David Petry]

On Wednesday, July 15, 2015 at 3:57:23 AM UTC-7, Jim Burns wrote:
> On 7/14/2015 12:29 AM, David Petry wrote:

> _If_ the minimal inductive set existed, _would_ it be a
> potential infinity?

Here's what I think about the word "exists". Computation "exists". It's very useful to think of the computer (i.e. an object that exists in the real world) as the mathematicians' microscope which helps mathematicians peer deeply into a world of computation. The mathematical objects that "exist" are things that can be observed using the computer as a microscope. The questions that can be answered (i.e. things that can be observed) in the world of computation are questions of the form "Does Turing machine T halt within (integer) N steps". Ultimately, though it's not trivial to do so, all of mathematics can be built upon questions of that kind. That's the foundation of the mathematics that can help us reason about the physical world.

Given that notion of existence, your question makes no sense. Or at least, I don't see how to make sense of it.


> This seems like a very straightforward,
> relevant question, with a very easy, short answer, for someone
> (like you) who finds all this so very easy.

Just to have a moment of fun here, the very easy short answer is "d'oh!".


> You wrote a couple paragraphs a few
> posts back about your motives, and seems that it is the undefined
> nature of infinity that gets you going.

That doesn't sound like anything I would write.

Motivation: As I see it, if we accept the idea that mathematics is a formalization of quantitative reasoning, then the foundations of mathematics would serve as a foundation for artificial intelligence. And I thought it would be really cool to pursue that idea further. But the foundations of mathematics as it is currently conceived by the pure mathematicians is missing a few things that it would need if it is to serve as a complete formalization of quantitative reasoning, the most salient of which is the principle of falsifiability.

The pure mathematicians often tell me that not only would it be unthinkable to accept falsifiability as a fundamental principle of mathematics, but it would be "insane" (their word!) to do so. The reason they can't accept it is because it would invalidate their beloved theory of infinity. And that's why I focus on infinity in these sci.math discussions.


> But infinity is not
> undefined, not in ZFC at least, which I think is the gold standard
> for mathematical-foundation-talk currently. Sets are undefined,
> is-an-element-of is undefined, and I think that's it. Everything
> else is defined from those primitive concepts. Infinity
> is not primitive in ZFC.

The really big problem here is the undefined notion of "is-an-element-of" when it comes to infinite sets. Here's a relevant quote from Hermann Weyl:

"... classical logic was abstracted from the mathematics of finite sets and their subsets .... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ...."

[Virgil]

In article <bb56f5e4-681f-4bd7...@googlegroups.com>,

Try to find a set of all natural numbers which does not defy any and
every definition of finiteness of a set. FAIL!

Find any standard set theory which does not have something like such a
set of all natural numbers. FAIl!

[Virgil]

In article <9a5efc5e-301c-4276...@googlegroups.com>,

It will belong to infinitely many different finite sets, and to their
union, which will NOT be finite!

[Virgil]

In article <a62bd70a-794c-489a...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Mittwoch, 15. Juli 2015 16:40:17 UTC+2 schrieb Virgil:
> > In article <797b5a96-4886-4b78...@googlegroups.com>,
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> >
> > > Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:
> > >
> > >
> > > > > > a mere digit sequence
> > > > > > can only define a natural number. One needs at least one radix
> > > > > > point or
> > > > > > one negative sign to name any other numbers!
> > > > >
> > > > > For an irrational number one needs more than available.
> > > >
> > > > In binary, one only needs two different digits for any real number!
> > >
> > > Then define an irrational number of your choice by digits. Remember:
> > > "LIM"
> > > and "SUM" and "..." are not digits.
> >
> > One can certainly define an actually infinite SEQUENCE of decimal digits
> > with decimal point by using "Sum"!
>
> But one cannot define an irrational number by merely listing digits.

WM snipped the one I previosuly defined that way.

So here is another: a binary point followed by an infinite sequence of
0's and 1's with 1's in and only in the digits in prime positions and
0's in composite positions.

[Jim Burns]

On 7/15/2015 7:20 PM, David Petry wrote:
> On Wednesday, July 15, 2015 at 3:57:23 AM UTC-7, Jim Burns wrote:
>> On 7/14/2015 12:29 AM, David Petry wrote:

>> _If_ the minimal inductive set existed, _would_ it be a
>> potential infinity?
>
> Here's what I think about the word "exists".
> Computation "exists". It's very useful to think of the computer
> (i.e. an object that exists in the real world) as the
> mathematicians' microscope which helps mathematicians peer deeply
> into a world of computation. The mathematical objects that "exist"
> are things that can be observed using the computer as a
> microscope.

Then the minimal inductive set exists in the same sense that 2
exists. The proof of the existence of the minimal inductive set
is as much of a calculation as 2 + 2 = 4, in every sense
relevant to this discussion.

Your computer-microscope does not "see" 2. It "sees" a
representation of 2. It can also "see" a representation of
the minimal inductive set. I have pointed you to the sort of
software that can do this.
http://www.wolframalpha.com/examples/Math.html

And so on.

> The questions that can be answered (i.e. things that
> can be observed) in the world of computation are questions of the
> form "Does Turing machine T halt within (integer) N steps".
> Ultimately, though it's not trivial to do so, all of mathematics
> can be built upon questions of that kind. That's the foundation
> of the mathematics that can help us reason about the physical world.
>
> Given that notion of existence, your question makes no sense.
> Or at least, I don't see how to make sense of it.

[...]

>> You wrote a couple paragraphs a few
>> posts back about your motives, and seems that it is the undefined
>> nature of infinity that gets you going.
>
> That doesn't sound like anything I would write.

These are the paragraphs I was thinking of:
<q>
>
> There's a reality out there. It's what we can observe and
> interact with. We can understand it, and we can have a shared
> understanding of that reality. Mathematics is a language that
> can help us understand that reality, and help us communicate
> with each other based on our shared understanding of that
> reality. If we don't establish that shared understanding of
> reality, we will be fighting forever. That's something I
> don't want.
>
> So as I see it, what you and the Cantorian mathematicians are
> doing is playing a game in which you can prove your own
> intellectual superiority, or something like that. And you keep
> trying to draw me into your game. And I hope you fail at drawing
> me into your game. I think your game will lead to endless
> fighting. I also think your game is pulling the whole world
> down into endless fighting. I'd like to crush your game out
> of existence.
>
</q>
<aef77c7d-856f-4ffe...@googlegroups.com>

>
> Motivation: As I see it, if we accept the idea that mathematics
> is a formalization of quantitative reasoning, then the foundations
> of mathematics would serve as a foundation for artificial
> intelligence. And I thought it would be really cool to pursue
> that idea further. But the foundations of mathematics as it is
> currently conceived by the pure mathematicians is missing a few
> things that it would need if it is to serve as a complete
> formalization of quantitative reasoning, the most salient of which
> is the principle of falsifiability.
>
> The pure mathematicians often tell me that not only would it be
> unthinkable to accept falsifiability as a fundamental principle
> of mathematics, but it would be "insane" (their word!) to do so.
> The reason they can't accept it is because it would invalidate
> their beloved theory of infinity. And that's why I focus on
> infinity in these sci.math discussions.
>
>

[...]

[David Petry]

On Wednesday, July 15, 2015 at 5:45:47 PM UTC-7, Jim Burns wrote:

> Then the minimal inductive set exists in the same sense that 2
> exists.

If we see mathematics as a framework for quantitative reasoning, then we are forced to accept '2' as an object that exists in mathematics; it has an essential role to play in quantitative reasoning.

You haven't given a convincing argument that the minimal inductive set has an essential role to play in quantitative reasoning.

[David Petry]

On Wednesday, July 15, 2015 at 8:35:07 AM UTC-7, Alan Smaill wrote:
> David Petry <david...@gmail.com> writes:


> > The constructivists have done a pretty good job of showing all, or
> > virtually all, of the mathematics that helps us reason about the real
> > world (e.g. physics) can be done constructively (i.e. using only the
> > notion of a potential infinity). It's very debateable whether I
> > should have written "or virtually all".

>
> Which constructivists are you thinking about?

Errett Bishop and Douglas Bridges, for starters.

[K_h]

"Ross A. Finlayson" wrote in message
news:1df5305b-886e-4c5a...@googlegroups.com...
>
> of them. (The Universe is
> infinite.)

No. The size of the physical cosmos is not known.

> The universe of mathematical
> objects would be its own
> powerset.

No, such a universe isn't a set at all, let alone a power set. The universe
of mathematical objects is a proper class.

> Transfinite cardinals contribute
> no analytical character to the
> reals, but infinitesimals do, and
> these are also fundamental features
> of parastatistics.

There are no infinitesimals on the real number line. Non-standard number
lines, for example the hyper-real number line, does have them.

> Yer finite math is weak.

Finite math is a very important branch of mathematics.

Best

[Virgil]

In article <98747e21-202f-4695...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Mittwoch, 15. Juli 2015 16:49:27 UTC+2 schrieb Virgil:
> > On Wednesday, July 15, 2015 at 12:10:30 PM UTC+1, WM wrote:
> > > Am Dienstag, 14. Juli 2015 23:04:10 UTC+2 schrieb Virgil:
> > <snip>
> > > > In binary, one only needs two different digits for any real number!
> > >
> > > Then define an irrational number of your choice by digits.
> >

> > A radix point followed by a sequence of binary digits which are 1 at and
> > only at an n!-th position following that radix point and zero elsewhere.

>
> n! is not a digit.

For each n in |N, n! is a naturally numbered digit position, which
collectively is enough everywhere outside of WM's worthless world of
WMytheology to successfully define an irrational number in any allowable
base!

[Virgil]

In article <2fa60adb-a2c6-4b7f...@googlegroups.com>,

It is finite formula successfully defining all infinitely many digits of
an infinite digit sequence representing an irrational number!
So, as usual, WM is totally wrong!

[Virgil]

David Petry <david...@gmail.com> writes:

> The constructivists have done a pretty good job of showing all, or
> virtually all, of the mathematics that helps us reason about the real
> world (e.g. physics) can be done constructively (i.e. using only the
> notion of a potential infinity).

Since corresponding proofs are almost always exist but tend to be much
longer and more difficult both to construct and to understand when done
"contructively" and few, if any, important old theorems fail to be still
provable and few, if any, interesting new theorems result, why bother?

[Virgil]

In article <1290c646-325c-4262...@googlegroups.com>,

David Petry <david...@gmail.com> wrote:

> On Wednesday, July 15, 2015 at 3:57:23 AM UTC-7, Jim Burns wrote:
> > On 7/14/2015 12:29 AM, David Petry wrote:
>
> > _If_ the minimal inductive set existed, _would_ it be a
> > potential infinity?
>
> Here's what I think about the word "exists". Computation "exists". It's
> very useful to think of the computer (i.e. an object that exists in the real
> world) as the mathematicians' microscope which helps mathematicians peer
> deeply into a world of computation. The mathematical objects that "exist"
> are things that can be observed using the computer as a microscope.

Since you would limit what you would allow to be seen to what can be
seen by a necessarily finite viewer, a computer, you are begging the
question.

[Virgil]

In article <4db21306-433a-451a...@googlegroups.com>,

You have not given a convincing argument that mathematics need be
restricted to your own peduliar forms of quantitative reasoning to the
exclusion of all its previously accepable forms of qualitative reasoning.

[WM]

Am Donnerstag, 16. Juli 2015 08:37:40 UTC+2 schrieb Virgil:
> In article <1290c646-325c-4262...@googlegroups.com>,
> David Petry <david...@gmail.com> wrote:
>
> > On Wednesday, July 15, 2015 at 3:57:23 AM UTC-7, Jim Burns wrote:
> > > On 7/14/2015 12:29 AM, David Petry wrote:
> >
> > > _If_ the minimal inductive set existed, _would_ it be a
> > > potential infinity?
> >
> > Here's what I think about the word "exists". Computation "exists". It's
> > very useful to think of the computer (i.e. an object that exists in the real
> > world) as the mathematicians' microscope which helps mathematicians peer
> > deeply into a world of computation. The mathematical objects that "exist"
> > are things that can be observed using the computer as a microscope.
>
> Since you would limit what you would allow to be seen to what can be
> seen by a necessarily finite viewer, a computer,

or your brain

> you are begging the
> question.

Regards, WM

[WM]

Am Donnerstag, 16. Juli 2015 05:06:46 UTC+2 schrieb K_h:
> "Ross A. Finlayson" wrote in message
> news:1df5305b-886e-4c5a...@googlegroups.com...
> >
> > of them. (The Universe is
> > infinite.)
>
> No. The size of the physical cosmos is not known.

The size of the cosmos as far at it is accessible and can be utilized for calculating purposes and doing mathematics, is finite and very small. You cannot even represent sll nuber between 1 and 10^10^100.

Regards, WM

[WM]

Am Donnerstag, 16. Juli 2015 02:45:47 UTC+2 schrieb Jim Burns:


> Your computer-microscope does not "see" 2. It "sees" a
> representation of 2.

There is nothing else but representations of 2. If you don't believe it, then try to find more.

> It can also "see" a representation of
> the minimal inductive set.

You can see a word that names the set or some axioms that define it, but you can never see more than a finite number of elements (or representations of elements).

Regards, WM

[Virgil]

In article <dfce5132-b1ae-48bf...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Donnerstag, 16. Juli 2015 02:45:47 UTC+2 schrieb Jim Burns:
>
>
> > Your computer-microscope does not "see" 2. It "sees" a
> > representation of 2.
>
> There is nothing else but representations of 2. If you don't believe it, then
> try to find more.

Then no such thing as the number that "2" represents has any physical
reality. Two and all other numbers exist only in the imagination!

>
> > It can also "see" a representation of
> > the minimal inductive set.
>
> You can see a word that names the set or some axioms that define it, but you
> can never see more than a finite number

One can physically see numerals but not physically see numbers!
One can only imagine numbers!

[Virgil]

In article <64af380b-8ba3-4551...@googlegroups.com>,

Eyes can see.
Brains, other than through eyes, cannot!
But brains can imagine!

Mathematics is a world of pure imagination. While what it imagines is
often, even usually, suggested by the physical world, it is not actually
of the physical world.

[abu.ku...@gmail.com]

I was thinking about using a base_pi,
which would use the three intefers,
1, 0, bar-2

[Jim Burns]

On 7/15/2015 10:48 PM, David Petry wrote:
> On Wednesday, July 15, 2015 at 5:45:47 PM UTC-7, Jim Burns wrote:

>> Then the minimal inductive set exists in the same sense
>> that 2 exists.
>
> If we see mathematics as a framework for quantitative reasoning,
> then we are forced to accept '2' as an object that exists in
> mathematics; it has an essential role to play in quantitative
> reasoning.

The _role_ that either 2 or the minimal inductive set _plays_
is not a _mathematical_ aspect of either mathematical object.

I asked you for a _mathematical_ definition of potential infinity
and actual infinity. If your reasons for letting 2 into your
universe and keeping the minimal inductive set out _must_
involve non-mathematical considerations (as certainly looks
like the case so far) then _you do not have a mathematical definition_ .

Ask yourself whether you are doing mathematics, or philosophy,
or something else.

_Are there_ mathematical definitions of potential infinity
and actual infinity _as you mean them_?

If there are, please give them. Without them, arguing with you
is like trying to lasso a column of smoke.

>
> You haven't given a convincing argument that the minimal
> inductive set has an essential role to play in quantitative
> reasoning.
>

The minimal inductive set absolutely _can_ be used in quantitative
reasoning. It is the standard model for the natural numbers.
So, the question is what you mean by "essential".

If by "essential" you mean it can't be replaced by an alternate
mathematical object, then I would say that the minimal inductive
set is not essential -- but by that standard whatever mathematical
object you use to represent 2 (perhaps { {}, {{}} }) is _also_
not essential.

What do you mean by "essential"?

By the way, some other quantitative uses for the minimal inductive set
(call it N):

From N and set theory we can construct Z without mentioning infinity.
From Z and set theory we can construct Q without mentioning infinity.
From Q and set theory we can construct R without mentioning infinity.

I don't know if it's a thing one can prove, but I suspect strongly
that the real numbers R are _essential_ to quantitative reasoning
about any continuum.

[abu.ku...@gmail.com]

that is to say that bar-over 2 is a 3-adic representation
of minus one

> which would use the three integers,
> 1, 0, bar-2

anyway, three is no longer the surfer's value of pi;
hasn't been since at least fifty years

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Wednesday, July 15, 2015 at 5:35:07 PM UTC+1, Alan Smaill wrote:
>> Julio Di Egidio <j***@diegidio.name> writes:
>> > On Wednesday, July 15, 2015 at 4:33:27 PM UTC+1, Virgil wrote:
>> >> In article <aa21dc63-5a88-4397...@googlegroups.com>,
>> >> Julio Di Egidio <j***@diegidio.name> wrote:
>> >>
>> >> > The confusion only exists thanks to standard theory: of course an
>> >> > inductive sequence is potentially infinite
>> >>
>> >> Any inductive sequence whose terms are regarded as forming a set
>> >> must form an actually non-finite set.
>> >
>> > For the umpteenth time: An standard sequence, inductive or otherwise,
>> > just cannot form a set. You need to take the limit of that sequence,
>> > then the set contains the limit as well.
>>
>> I agree that there is an important difference between a generating
>> process and any potential or actual collection of generated elements.
>
> That is patent nonsense, thank you.

What exactly is nonsense? (What does "Taht" refer to)

> For the umpteenth time: a never
> finished gathering process would be a potentially infinite collection,

You think the (never finished) process *is* the potentially infinite
collection?

> and that is different from an actually infinite collection, i.e. the
> putatively completed process. In particular, a standard inductive
> sequence is a potential infinity, an infinite set is an actual
> infinity.

You see no difference between a (single) generating process, and the
elements generated? (I am trying to be sure I follow your claims)

>> But what do you mean by the "limit of the sequence", and why is it
>> needed?
>
> Because that is all we have, structural limits.

I don't know what the "structural limit" of a sequence is, in general.
Notions of limit usually involve a pre-existing notion of distance
(or topology), but that is not apparent for a general sequence.
Can you say what a structural limit for sequences is?


>
> Julio

--
Alan Smaill

[Alan Smaill]

WM <wolfgang.m...@hs-augsburg.de> writes:

[when asked for the second time]

> Am Mittwoch, 15. Juli 2015 17:40:07 UTC+2 schrieb Alan Smaill:
>
>
>> Accepting the set of natural numbers as a completed
>> entity, is already an example of actual infinity, isn't it?
>>
> It is paying a lip service.

So, it *is* an example of actual infinity.

Why do you find it so uncomfortable to admit this, I wonder?

>
> Regards, WM

--
Alan Smaill

[abu.ku...@gmail.com]

in base_ten, it is the equivalent of three's compliment
(subtract from ...0000.0, and
two's compliment (subtract from ...9999.0

[WM]

If you say that |N is actually infinite, then the consequence is that it has a fixed number, call it aleph_0, of elements.

If you say it is actually infinite without this consequence, then you talk nonsense.

Regards, WM

[Virgil]

In article <e0ce0006-f37b-4ae0...@googlegroups.com>,

WM <wolfgang.m...@hs-augsburg.de> wrote:

> Am Freitag, 17. Juli 2015 17:40:07 UTC+2 schrieb Alan Smaill:
> > WM <wolfgang.m...@hs-augsburg.de> writes:
> >
> > [when asked for the second time]
> >
> > > Am Mittwoch, 15. Juli 2015 17:40:07 UTC+2 schrieb Alan Smaill:
> > >
> > >
> > >> Accepting the set of natural numbers as a completed
> > >> entity, is already an example of actual infinity, isn't it?
> > >>
> > > It is paying a lip service.
> >
> > So, it *is* an example of actual infinity.
> >
> > Why do you find it so uncomfortable to admit this, I wonder?
>
> If you say that |N is actually infinite, then the consequence is that it has
> a fixed number, call it aleph_0, of elements.

To say that a set has infinitely many elements doesnNOT mean that it has
any particular number of elements unless one assumes that there can be
only one size of infinite set, which, at least outside of WM's worthless
world of WMytheology, need not be assumed.

>
> If you say it is actually infinite without this consequence

Then you re not making unprovable assumptions.

[Dan Christensen]

You must be working with some seriously weird definition of infinite, WM. (What else is new?) By a set S being infinite, I just mean that there exists an injective function f: S --> S that is not surjective. There is no maximum element as you suggest. If there was, it wouldn't be infinite! I make no formal distinction between actual or potential infinity. AFAIK, no mathematicians do.

BTW, you never did tell us your own formal definition of finite or infinite -- actual, potential or whatever. What seems to be the problem, WM? Don't like to be pinned down to one definition? Too, ummm... restrictive?

Dan

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