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Set Theory: Should you believe?
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DontBother@nowhere.net
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Joined: 13 Jul 2006
Posts: 114

PostPosted: Thu Jul 20, 2006 3:31 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On Wed, 19 Jul 2006 19:47:02 -0600, Virgil <virgil@comcast.net> wrote:

Quote:
In article <tvbtb29qjsde0k18m272crf3092esavcsu@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 14:47:00 -0600, Virgil <virgil@comcast.net> wrote:

You have not read my reference to " logical tautologies" correctly.

If, for example. "P and not P" would qualify as "false" does your
gobledegook require its negation, "P or not P", to be true?

"P and not P" is only universally false because it provides no
mechanical basis for alternatives since any "not (P and not P)"
converts into itself "not P and P".

Not in any respectable logic it doesn't. According to de Morgan's laws,
"not (P and not P)" is logically equivalent to "P or not P".

It may be equivalent to lots of things. The issue is whether it
converts into itself mechanically.

Quote:
So there is no alternative in
strict mechanical terms because "P or not P" is not always true if P
itself contains a self contradiction because P and "not P" are the
same.

If P contains a contradiction, doesn't that make P false? In normal
logic, that would make (not P) true.

It makes them the same.

Quote:
See if the above impresses you even a little.

Not even.


Good. Then moving right along.

~v~~
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Dave L. Renfro
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PostPosted: Thu Jul 20, 2006 3:39 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Dave L. Renfro wrote:

Quote:
What do you mean by the truth of an axiom?

Lester Zick wrote:

Quote:
Whether an axiom is true or not.

Axioms are true by definition. Of course,
a certain axiom A for Theory T may be false
in Theory T'. However, in any (consistent)
theory for which A is an axiom, A will be true.

If you wish to argue with me further, please
do so with specific references from specific
mathematical logic and/or mathematical foundations
texts that support what you're saying.

Dave L. Renfro wrote:

Quote:
Not only that, but axioms can be proved
quite easily. Here's an example I posted
back on June 20:

Axiom R: All right angles are congruent.

Theorem: All right angles are congruent.

Proof (2-column format):

Statements Reasons

1. All right angles are congruent. 1. Axiom R.

Lester Zick wrote:

Quote:
You mean mathematikers can rely on circular
reasoning to demonstrate modern math theorems?
How nice. Certainly supports every speculative
conjecture I've made concerning the intellectual
content of modern math. Don't prove it; just
assume it; then claim you've proven it.

Mathematics is basically the study of various
deductive structures. It appears that you
think it's something else.

One *can* study structures generated from
P ==> P for various statements P, but most
people aren't going to be very interested.

Dave L. Renfro
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Aatu Koskensilta
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PostPosted: Thu Jul 20, 2006 3:48 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Lee Rudolph wrote:
Quote:
Aatu Koskensilta <aatu.koskensilta@xortec.fi> writes:

My remarks are based on somewhat more mundane considerations; in group
theory one often says things like "let A be an Abelian group such that
..." but in number theory one does not say "let <W,s,f,g> be a structure
in which all the arithmetical consequences of ZFC hold" or "let
W,s,f,g> be a natural number structure in which the twin prime
conjecture holds".

I agree (based on observations of number theorists) that "one does
not say" that sort of thing, and I am open to being persuaded (indeed,
I am predisposed to be persuaded) that if a school of mathematicians
got into the habit of saying that sort of thing (while continuing to
do mathematics) then we (and possibly they) might want to say that
what they were doing (though still mathematics, and possibly very
fine mathematics) was no longer "number theory" (or, weaker, no
longer *just* "number theory"): but I don't see such a response
is self-evidently right.

That certainly would be the natural reaction. In fact, there already is
a discipline of mathematics where one can expect to hear such things. No
one calls it "number theory". It seems highly unlikely that there ever
could be a school of mathematics in which studying e.g. structures in
which the arithmetical consequences of ZFC hold was called "number
theory". That sort of a terminological shift would require a major
upheaval in the way people think about natural numbers.

Quote:
In other words (I guess), is there anything more than historical
chance and prejudice behind the feeling (which I certainly share,
but don't feel particularly justified in sharing) that "natural
numbers should be *categorical*, dammit"?

It's a basic property of our conception of the natural numbers that they
don't bifurcate into a multitude of non-isomorphic structures. There are
some people of ultra-intuitionist and ultra-finitist peruasion -
Esenin-Volpin and Edward Nelson come to mind - who do think that there
are many different natural number lines and reject the ordinary
conception of the natural numbers as incoherent or unjustified. This
just goes to show, once again, that it is to a large extent a matter of
personal preference and inclination what one finds convincing and
coherent, and that even in mathematics there probably is no principle
someone competent hasn't doubted or rejected.

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Dave L. Renfro
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PostPosted: Thu Jul 20, 2006 3:50 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Dave L. Renfro wrote (in part):

Quote:
Axioms are true by definition. Of course,
a certain axiom A for Theory T may be false
in Theory T'. However, in any (consistent)
theory for which A is an axiom, A will be true.

Actually, I think "truth" only makes sense
once a model is specified, so "for Theory T"
should be "in Model M for Theory T", and
similarly for the other references to "theory"
above.

Dave L. Renfro
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DontBother@nowhere.net
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PostPosted: Thu Jul 20, 2006 4:15 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On Wed, 19 Jul 2006 21:32:14 -0600, Virgil <virgil@comcast.net> wrote:

Quote:
In article <48ptb2h9tn62b5qq2hifgras3vakbotcnn@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

A good beginning discussion of the subject of mathematical definitions
is in Suppes's 'Introduction To Logic'. But in order not to inhibit the
metastasis of your own convictions, I recommmend that you not read such
books.

Well if there's one thing I detest more than assumptions of truth it's
metastasis of convictions when one is actually dealing instead with
demonstrations of truth.

As Zick has not demonstrated any truths

Neither have you, sport. All we've dealt with so far is set theory as
a faith based institution of doddering ineptitude.

Quote:
nor dealt with any,...

~v~~
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DontBother@nowhere.net
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PostPosted: Thu Jul 20, 2006 4:26 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On 20 Jul 2006 02:18:57 -0700, "Rupert" <rupertmccallum@yahoo.com>
wrote:

Quote:

Lester Zick wrote:
On 18 Jul 2006 21:52:25 -0700, "Rupert" <rupertmccallum@yahoo.com
wrote:


Lester Zick wrote:
On 18 Jul 2006 18:19:16 -0700, "Rupert" <rupertmccallum@yahoo.com
wrote:


Nam Nguyen wrote:
Lester Zick wrote:
On Tue, 18 Jul 2006 07:03:50 GMT, Nam Nguyen <namducnguyen@shaw.ca
wrote:



Virgil wrote:

In article <q36ob213q98jmnaddpjfa95lk5ms282krq@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:



The question I have is whether you or others believe in the
possibility of universally exhaustively true mathematical axioms?


What is "truth"?

I can deal with the tautologous logical truth of implications like "if P
then (P or Q)", but other than those, which include the more complex
logical deductions from a set of axioms, I know of no absolute truth.

If we care to consider absolute truth, then there is no such
thing as an absolute truth.


Well thanks for the input. Can we take your word for this?

I don't see why not, unless an absolute truth could be demonstrated
to exist! Would

(1) (P => (P \/ Q))

constitute an absolute truth? Well, at this moment for some odd reason
"\/" to me means what "/\" means to a lot of people (and vice versa for
"/\")!. So (1) to me is not a truth; so it can not be an absolute
truth that *must be universally recognized without exception*!


If "absolute truth" means "something that is true regardless of which
semantics you use" then you're right, no absolute truths exist. This is
not a very interesting point.

I agree it's not a very interesting point since you assume this truth
absolute.


No, I don't. I certainly don't assume it absolute in Nam Nguyen's
sense. Clearly no truths are absolute in that sense.

Oho?

What I said was true relative to the semantics I was actually using. It
clearly would not be true relative to any conceivable semantics. This
is trivial.

Certainly it seems so to those prefer to deal in assumptions of truth
and assume so.


No "seems" about it. It's completely obvious. What contestable
assumption do you think I'm making?

I didn't say "contestable assumption". What I said was "assumption".
Assumptions refer to a lack of demonstration. You certainly assume
what you don't demonstrate by calling it obvious. Obviously it is a
canonical assumption on your part that what you say is true. Whether
it is actually true or not however remains to be seen.

Quote:
I think people who talk about absolute truth usually mean something
different to what Nam Nguyen thinks it means

Not at all

What's your evidence?

If you would be so good as to clarify what Nam thinks "absolute truth"
means I'll try to supply it. It may or may not be "different to" what
others think but I have no doubt it will be "different from" what you,
Nam, and others think.

Quote:
if I intuit your point correctly. The claim I make is
intended literally and exactly.


What claim?

What claim were you referring to? We seem to be in a "he said/she
said" mode at this point where it's difficult to discern what each of
us is referring to. My claim in general terms is that the alternative
to absolute falseness is universally true.

Quote:
, and I make no comment
about whether absolute truth exists in on of these senses or whether
what I said was absolutely true in one of these senses. I merely claim
that it was true.

That is: not even "tautologous logical
truth" would be absolute.


Yes but tautological alternatives to necessary and universal falseness
would perforce have to be necessarily and universally true.

Again, as has been questioned by another poster, what does "universal
falseness", or "universally true" mean?

~v~~

--
-----------------------------------------------------

What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
----------------------------------------------------

~v~~

~v~~

~v~~
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PostPosted: Thu Jul 20, 2006 4:36 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On Thu, 20 Jul 2006 11:39:47 EDT, "Dave L. Renfro"
<renfr1dl@cmich.edu> wrote:

Quote:
Dave L. Renfro wrote:

What do you mean by the truth of an axiom?

Lester Zick wrote:

Whether an axiom is true or not.

Axioms are true by definition.

Not quite. Axioms are true by assumption. That doesn't make them true.
It is one of the quirks of moderm mathematics that the meaning of
"definition" has been converted to "arbitrary assertion" or "fiat"
solely in order to validate assumptions of truth without coming right
out and saying so. Truth by definition and assumptions of truth are
nothing more than revealed truth, divine truth and are purely faith
based and undemonstrated intimations of intuition.

Quote:
Of course,
a certain axiom A for Theory T may be false
in Theory T'. However, in any (consistent)
theory for which A is an axiom, A will be true.

Which is just another way of saying axioms can be false with a
straight face.

Quote:
If you wish to argue with me further, please
do so with specific references from specific
mathematical logic and/or mathematical foundations
texts that support what you're saying.

Why would I want to argue with you? You haven't said anything worth
arguing about.

Quote:
Dave L. Renfro wrote:

Not only that, but axioms can be proved
quite easily. Here's an example I posted
back on June 20:

Axiom R: All right angles are congruent.

Theorem: All right angles are congruent.

Proof (2-column format):

Statements Reasons

1. All right angles are congruent. 1. Axiom R.

Lester Zick wrote:

You mean mathematikers can rely on circular
reasoning to demonstrate modern math theorems?
How nice. Certainly supports every speculative
conjecture I've made concerning the intellectual
content of modern math. Don't prove it; just
assume it; then claim you've proven it.

Mathematics is basically the study of various
deductive structures. It appears that you
think it's something else.

One *can* study structures generated from
P ==> P for various statements P, but most
people aren't going to be very interested.

Dave L. Renfro

~v~~
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DontBother@nowhere.net
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PostPosted: Thu Jul 20, 2006 4:46 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On Thu, 20 Jul 2006 11:50:15 EDT, "Dave L. Renfro"
<renfr1dl@cmich.edu> wrote:

Quote:
Dave L. Renfro wrote (in part):

Axioms are true by definition. Of course,
a certain axiom A for Theory T may be false
in Theory T'. However, in any (consistent)
theory for which A is an axiom, A will be true.

Actually, I think "truth" only makes sense
once a model is specified, so "for Theory T"
should be "in Model M for Theory T", and
similarly for the other references to "theory"
above.

And into what model do "model M and theory T" fall precisely?

Modern appeals to model theory and models are simply naive
alternatives to being able to demonstrate what's actually true and
false. Models are not mechanically exhaustive. That's why they're
models of what we suppose to be true and not what's actually
demonstrably true and appeals to models instead of truth merely
demonstrate the rankest kind of faith based mathematics.

~v~~
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Virgil
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Posts: 5536

PostPosted: Thu Jul 20, 2006 6:37 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <bm7vb2t2arba9mrpdujjjnp8bg0k5pc58s@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Wed, 19 Jul 2006 20:07:44 -0600, Virgil <virgil@comcast.net> wrote:

In article <ocntb2p6ahgltjflpf9ed7b8dcqgl2ve6n@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

But where does one find this alleged absolute falseness? Does Zeck have
any concrete examples of this mysterious substance?

I'm not familiar with what Zeck has. What Zick has is exhaustive
alternatives neither of which you seem willing to acknowledge is true
or false but prefer to demand examples of things which are true or
false instead so you can argue about their truth or falsity instead of
universally true and false alternatives in general.

Apparently Zick has nothing that Zeck does not have.

Zick has presented no "exhaustive alternatives", he merely keeps talking
as if there were some.

And Verge keeps talking as if there were none.

I am talking as if it has not ben established whether there are any, at
least until one has assumed something on which to base distinguishing
alternatives.
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Virgil
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PostPosted: Thu Jul 20, 2006 6:40 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <bo7vb2pc405sis5hkqn05phm7jiut84v6j@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Wed, 19 Jul 2006 20:11:06 -0600, Virgil <virgil@comcast.net> wrote:

In article <8intb2dm8drmqqbih0p19tmgbcjunep3vk@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:


"Not" is the tautological alternative to "not not". The excluded
middle is the reason we must reduce possible predicates to an absolute
mechanical minimum.

Then where do "not not not" and "not not not not", and so on, fit in?

Do "not" and "not not" exhaustive possibilities for truth?

Not to me.

Then what does?

Zick is the one making claims about them, I merely express doubt.

Zick has yet to exhibit any "exhaustive alternatives" that do not
require presuming anything.
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Virgil
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PostPosted: Thu Jul 20, 2006 6:49 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <pr7vb2h029h5cttph7cdapmb8ugon172rc@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Wed, 19 Jul 2006 20:00:59 -0600, Virgil <virgil@comcast.net> wrote:

In article <tomtb2la22cfo3digictmdgp6bnuh38dcr@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 17:52:29 -0600, Virgil <virgil@comcast.net> wrote:

In article <itatb21s91un4gdkr6ekcmrei00h0ju97f@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 12:33:51 -0600, Virgil <virgil@comcast.net> wrote:

If you are unable to provide any examples of what you say you mean, I
take leave to doubt that you know what you mean, or that you mean
anything at all.

Who says I'm unable

I did not say you could not, only that you have not.

Nonsense. You said "unable". If you can't even recognize your own
words there's nothing to talk about.

I said "if you are unable".

Which you denied you said.

In Zick's dreams.

I have not denied saying "if Zick is impotent" in this issue but only
Zick misrepresents that as me claiming that Zick is impotent.
Quote:

If Zick cannot recognize the difference, perhaps he is the one who who
has nothing to talk about.

So far Zick has given no evidence of being able to give any examples,

And Verge has given no evidence of being able to reason about truth.

As Zick has given no evidence that he has any absolute truth, he has
shown nothing to reason about.


Quote:
As you have given me no reason to suppose that there are any such things
as 'absolute falseness' or 'universal truth', much less that you can
provide instances of either, I have good reason to doubt you.

I've given the example of alternatives to absolute falseness which you
cannot judge.

On the contrary, I can judge it garbage.
Quote:

My impression is that agreeing that "alternatives to universal
falseness would perforce be universally true" requires agreement on at
least what "universal falseness" means and what "universal truth" means,
and I have no referents for either.

I can't see how you need to understand what either means in order to
evaluate the general proposition provided you understand that they're
alternatives to one another.

You claim them to be. But on what evidence? Until you provide evidence
that at least one of them actually exists, there is nothing to discuss.

Uh not really when you can't judge the hypothetical.


If they are merely hypothetical, then you are asking to have them
assumed in a context that requires that nothing be assumed.
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PostPosted: Thu Jul 20, 2006 6:51 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <a48vb2d4qv9dg49v8nsjgcqjc9gbqg9cd1@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Wed, 19 Jul 2006 20:04:59 -0600, Virgil <virgil@comcast.net> wrote:

In article <3antb2hc9porstidi882cppdnmnmu3ef62@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 17:26:34 -0600, Virgil <virgil@comcast.net> wrote:

In article <dk8tb21ddntqu30j88mrckdr773ktuc4r0@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Tue, 18 Jul 2006 21:38:24 -0600, Virgil <virgil@comcast.net> wrote:

In article <4q8rb2t8c4q4g5hn5u1ht55h2pm6veb4gk@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:


Well, Nam, without going on endlessly I would like to ask if
arguments
based on "absolute truth" are indeed futile, are arguments based on
"absolute falseness" necessarily equally futile?

If one excludes logical tautologies like "if P then P" and logical
contradictions like "if P then not P", yes.

I had other tautologies in mind of the general form "P, not P". But
the general argument remains regardless. Universal alternatives to
universal falseness must perforce be universally true.

Provided that there are any of either, maybe. But I cannot assent
without seeing exemplars.

Sure you can. Or if you can't you shouldn't be doing mathematics. If
alternatives are exhaustive one or the other must be true.

Provided that any of the alleged alternatives exist at all and provided
that "true" versus "false" is appropriate for any of them. For example
"North", East" "South" and "West" are in a sense exhaustive, but none of
them is any truer that another.

Sure. And Verge and math are mutually exhaustive alternatives too.


If they both exhaust you, perhaps you should avoid them.
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Virgil
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PostPosted: Thu Jul 20, 2006 6:56 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <n68vb2p91g8m3nbbp0k4v7qbit7m6jfutn@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Wed, 19 Jul 2006 19:47:02 -0600, Virgil <virgil@comcast.net> wrote:

In article <tvbtb29qjsde0k18m272crf3092esavcsu@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

On Wed, 19 Jul 2006 14:47:00 -0600, Virgil <virgil@comcast.net> wrote:

You have not read my reference to " logical tautologies" correctly.

If, for example. "P and not P" would qualify as "false" does your
gobledegook require its negation, "P or not P", to be true?

"P and not P" is only universally false because it provides no
mechanical basis for alternatives since any "not (P and not P)"
converts into itself "not P and P".

Not in any respectable logic it doesn't. According to de Morgan's laws,
"not (P and not P)" is logically equivalent to "P or not P".

It may be equivalent to lots of things. The issue is whether it
converts into itself mechanically.

According to de Morgan, and others,
"not (P and not P)" and
"P or not P"
convert quite mechanically into each other but
"not (P and not P)" and
"not P and P"
do not convert into each other in any way at all.
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PostPosted: Thu Jul 20, 2006 7:02 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <utavb2tsrf1vme8aijr4a59bp5q450ks98@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
On Wed, 19 Jul 2006 21:32:14 -0600, Virgil <virgil@comcast.net> wrote:

In article <48ptb2h9tn62b5qq2hifgras3vakbotcnn@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

A good beginning discussion of the subject of mathematical definitions
is in Suppes's 'Introduction To Logic'. But in order not to inhibit the
metastasis of your own convictions, I recommmend that you not read such
books.

Well if there's one thing I detest more than assumptions of truth it's
metastasis of convictions when one is actually dealing instead with
demonstrations of truth.

As Zick has not demonstrated any truths

Neither have you, sport.

AS I am the one doubting the existence of any such things as absolute
truths or absolute falsehoods, my lack of demonstrating the existence of
any such thing supports my position.

As Zich is the one affirming their existence, his lack of demonstration
tends to weaken his position.

Quote:
All we've dealt with so far is set theory as
a faith based institution of doddering ineptitude.

Zick has illustrated doddering ineptitude by his own performance, but
that ineptitude has not been up much of anything else.
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PostPosted: Thu Jul 20, 2006 7:12 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <d1bvb29uf4ujvoib634ps0hqppje11iufa@4ax.com>,
Lester Zick <DontBother@nowhere.net> wrote:

Quote:
I didn't say "contestable assumption". What I said was "assumption".
Assumptions refer to a lack of demonstration. You certainly assume
what you don't demonstrate by calling it obvious.

In mathematics, all assumptions (axiom systems) are merely conditional,
to see what will follow from them. When what follows proves useful or
interesting, one tends to codify those assumptions. but that never
requires that one claims them true is any absolute sense. Such
assumptions are always "what if's".



Quote:
Obviously it is a
canonical assumption on your part that what you say is true. Whether
it is actually true or not however remains to be seen.

But it is something that Zilch is unable to see.

Quote:

My claim in general terms is that the alternative
to absolute falseness is universally true.

A claim which means nothing, unless Zilch can establish that there is an
absolute falseness. Is that anything like establishing that "Satan"
exists?
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