PROOF INFINITY DOES NOT EXIST! Not even ONE type [Logic]

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From: Wolf K on 12 Jul 2010 10:10

On 11/07/2010 12:16, George Greene wrote:
> On Jul 10, 12:31 pm, Wolf K<weki...(a)sympatico.ca> wrote:
>> (still dazed by the nonsense passed off as "grammar
>> in grade 6)
>
> No farting in church. This is sci.logic.
> Even if grammar isn't all that coherent for NATURAL languages, it is
> still very much
> necessary&relevant for computer programming languages, formal
> languages, and logic.
> Around HERE, what you call "nonsense" was a noble attempt to get
> people to think straight
> and write clearly -- even if it got a little Procrustean at times.

I taught a course in the history of English before I descended into the
real world and taught grammar in high school, Where I discovered that
much of what my students believed about English was nonsense, and had
been learned in middle school. What they had actually learned were some
(often poorly understood) rules about how to speak middle class English.

"Grammar" has a different meaning for artificial formal languages than
in natural languages. For a formal language, the grammar is simply the
definition of well-formed statements in that language; some also include
rules of inference, which is OK by me. Grammars of natural language are
contingent theories of how its speakers form utterances that "make
sense". The "grammar" taught in middle school is in large part a
description of a class dialect. Since to be successful one needs to
master this dialect, it is of course useful to teach it.

cheers,
wolf k.

From: Milton J. Smuthworthy, I on 12 Jul 2010 12:31

Then Don Stockbauer says:
>>>
>>> "|-|ercules" <radgray...(a)yahoo.com> wrote
>>>>> =A0To me, it is self-evident that the axiom of infinity is true.
>
>I have an affinity for infinity.

I'm finicky about these infinity foibles.

From: Don Stockbauer on 12 Jul 2010 12:51

On Jul 12, 11:31 am, "Milton J. Smuthworthy, I"
<tonworthyCLOT...(a)gmail.com> wrote:
> Then Don Stockbauer says:
>
>
>
> >>> "|-|ercules" <radgray...(a)yahoo.com> wrote
> >>>>> =A0To me, it is self-evident that the axiom of infinity is true.
>
> >I have an affinity for infinity.
>
> I'm finicky about these infinity foibles.

Coy bulls make poor breeders.

From: Dan Christensen on 12 Jul 2010 15:11

On Jul 10, 7:22 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote:
> On Jul 9, 10:17 am, Dan Christensen <Dan_Christen...(a)sympatico.ca>
> wrote:
>
> > I can't imagine that you would be able to do very much using
> > "finitist" methods. How do they handle such basic concepts as the
> > square root of 2?
>
> Terence Tao in "A computational perspective on set theory"http://terrytao..wordpress.com/2010/03/19/a-computational-perspective-...
>
> in which he explores the question "what is the finitary analogue of
> statements such as Cantors theorem or the Banach-Tarski paradox?"
> summarizes thus:
>
> <quote>
> The above discussion suggests that it is possible to retain much of
> the essential mathematical content of set theory without the need for
> explicitly dealing with large sets (such as uncountable sets), but
> there is a significant price to pay in doing so, namely that one has
> to deal with sets on a "virtual" or "incomplete" basis, rather than
> with the "completed infinities" that one is accustomed to in the
> standard modern framework of mathematics. Conceptually, this marks
> quite a different approach to mathematical objects, and assertions
> about such objects; such assertions are not simply true or false, but
> instead require a certain computational cost to be paid before their
> truth can be ascertained. This approach makes the mathematical
> reasoning process look rather strange compared to how it is usually
> presented, but I believe it is still a worthwhile exercise to try to
> translate mathematical arguments into this computational framework, as
> it illustrates how some parts of mathematics are in some sense "more
> infinitary" than others, in that they require a more infinite amount
> of computational power in order to model in this fashion. It also
> illustrates why we adopt the conveniences of infinite set theory in
> the first place; while it is technically possible to do mathematics
> without infinite sets, it can be significantly more tedious and
> painful to do so.
> </quote>

With your "countably infinite loops" (see link), it seems to you are
sneaking infinite sets in through the back door. You posit an
algorithm that can complete an infinite, countable number of
iterations (ranging over ALL the natural numbers) and arrive at some
conclusion. Have such notions ever been successfully formalized?

Dan
Download my DC Proof software at http://www.dcproof.com

From: Dan Christensen on 12 Jul 2010 17:24

Correction

On Jul 10, 7:22 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote:
> On Jul 9, 10:17 am, Dan Christensen <Dan_Christen...(a)sympatico.ca>
> wrote:
>
> > I can't imagine that you would be able to do very much using
> > "finitist" methods. How do they handle such basic concepts as the
> > square root of 2?
>
> Terence Tao in "A computational perspective on set theory"http://terrytao..wordpress.com/2010/03/19/a-computational-perspective-...
>
> in which he explores the question "what is the finitary analogue of
> statements such as Cantors theorem or the Banach-Tarski paradox?"
> summarizes thus:
>
> <quote>
> The above discussion suggests that it is possible to retain much of
> the essential mathematical content of set theory without the need for
> explicitly dealing with large sets (such as uncountable sets), but
> there is a significant price to pay in doing so, namely that one has
> to deal with sets on a "virtual" or "incomplete" basis, rather than
> with the "completed infinities" that one is accustomed to in the
> standard modern framework of mathematics. Conceptually, this marks
> quite a different approach to mathematical objects, and assertions
> about such objects; such assertions are not simply true or false, but
> instead require a certain computational cost to be paid before their
> truth can be ascertained. This approach makes the mathematical
> reasoning process look rather strange compared to how it is usually
> presented, but I believe it is still a worthwhile exercise to try to
> translate mathematical arguments into this computational framework, as
> it illustrates how some parts of mathematics are in some sense "more
> infinitary" than others, in that they require a more infinite amount
> of computational power in order to model in this fashion. It also
> illustrates why we adopt the conveniences of infinite set theory in
> the first place; while it is technically possible to do mathematics
> without infinite sets, it can be significantly more tedious and
> painful to do so.
> </quote>

With your "countably infinite loops" (see link), it seems you are
sneaking infinite sets in through the back door. You posit an
algorithm that can complete an infinite, countable number of
iterations (ranging over ALL the natural numbers) and arrive at some
conclusion. Have such notions ever been successfully formalized
without referring to the set of natural numbers as a whole?

Dan
Download my DC Proof software at http://www.dcproof.com