'When Are Relations Neither True Nor False? [Logic]

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From: James Burns on 26 Mar 2010 11:55

Transfer Principle wrote:
> I wonder how many posters in any poll (whether formal
> or informal) would vote for 0.999...=1 once they realize
> that 0.999... has _infinitely_ many nines, and so must
> differ by 1 by less than any measurable (as Burns points
> out here) amount.

If you're going to conduct another poll at some point,
consider asking the pollee to compare
0.99... and 0.9999...

> This discussion concerns the theory Z-Regularity and
> what it proves (as opposed to ZF-Regularity). Yet the
> proof of 0.999...=1 to which I linked explicitly lists the
> Replacement Schema as one of the axioms that is used
> in the proof.
>
> In trying to determine exactly where an instance of the
> Replacement Schema is used in the proof, it appears that
> the schema is used to construct the real numbers. It turns
> out that the proof that 1 is the multiplicative identity on N
> is the first proof that requires Replacement:

That you have a proof that /uses/ Replacement is not
the same thing as a proof /requiring/ Replacement.

The second is much harder to prove. Merely failing to
find a proof of 0.999... = 1 in ZF-Regularity is not
enough (unless you have first proved your omniscience).

Perhaps you could find a non-standard model of
ZF-Regularity in which 0.999... = 1 is false.
Better you than me, though. You would need to re-create
much of every-day mathematics, only with new and
highly un-intuitive interpretations of symbols and
sentences we learned practically at our mothers'
knees. I boggle.

[...]
> Of course, someone is likely to point out that this is due to
> my overreliance on free sites such as Metamath, and if I
> could afford an actual _textbook_, I'd learn exactly how to
> construct the real numbers without Replacement Schema.

"Of course, someone is likely to point out..."
It looks to me as though you like a good brawl so much
that, if no one else is interested, you'll just go
ahead without them.

I don't know if there are any textbooks out there that
construct the reals without using Regularity. I don't
know why an author would go very far out of their way
to do so, since they're supposedly using ZF(C), not
ZF-Regularity.

This thing about poor little you not being able to
afford a textbooks is at least as tiresome as that
other thing about you not being able to afford a newsreader.
You have one advantage, in that no one else really
knows what you can or cannot afford. On the other hand,
you don't seem to be aware of how inexpensive a perfectly
adequate math or logic textbook could be. My suspicion
is that you don't want to know. Your rhetorical use
of your lack of textbooks suggests that you don't want
to give up any opportunity to wag your finger at the
"exclusivity" of the world-wide community of mathematicians.

Here's your chance to prove me wrong though:
How much would you be willing to spend on a textbook?

Maybe you should do a little research before you answer,
or else you may end up with a perfectly fine textbook
that you have admitted you could afford. Check Amazon
for used textbooks. Check for downloadable texts, such
as Wikibooks or the Gutenberg Project.

And, if you answer (IF you answer), say, 10 cents,
including shipping and handling, then I think it will
be obvious just how serious poor little you are about
getting a logic or math textbook.

Jim Burns

From: MoeBlee on 26 Mar 2010 12:55

On Mar 26, 12:20 am, Transfer Principle <lwal...(a)lausd.net> wrote:

> the
> proof of 0.999...=1 to which I linked explicitly lists the
> Replacement Schema as one of the axioms that is used
> in the proof.

Replacement isn't needed to prove .999... = 1. With David Ullrich's
help, I posted a rigorous proof (in Z-regularity) of .999... = 1 in a
thread for that purpose a few years ago.

> In trying to determine exactly where an instance of the
> Replacement Schema is used in the proof, it appears that
> the schema is used to construct the real numbers.

Replacement is not needed to construct the real numbers. Either the
Dedekind cuts approach or the equivalence classes of Cauchy sequences
approach are just fine in Z-regularity.

> It turns
> out that the proof that 1 is the multiplicative identity on N
> is the first proof that requires Replacement:

That 1 is the multiplicative identity does not require replacement.

MoeBlee

From: MoeBlee on 26 Mar 2010 13:04

On Mar 26, 12:42 am, Transfer Principle <lwal...(a)lausd.net> wrote:

> as powerful

This cries for a definition or explanation of what is meant by 'as
powerful'. In mathematical logic we have various notions such as
'intepretability' and 'conservative extension'. But some of your
comments seem not to use 'as powerful' in such technical senses
(otherwise some of your questions in this regard would be non-
starters).

MoeBlee

From: Nam Nguyen on 26 Mar 2010 13:36

Daryl McCullough wrote:
> Nam Nguyen says...
>
>> Daryl McCullough wrote:
>>
>>> Ultimately, the people on this newsgroup who object to standard
>>> mathematics are really objecting to the idea that there can be
>>> such a thing as a counter-intuitive result.
>> That's a grossly erroneous over-generalization. It's many of those who
>> defend standard mathematics who erroneously object to the counter-intuitive
>> _but real nature_ of mathematics: relativity of truth and provability.
>
> Huh? Standard mathematics perfectly well takes into account the
> "relativity of truth". Truth is relative to an interpretation. So
> your objection makes no sense.

[Note that I mentioned "many of those" not "all"].

Glad that you and I agree "Truth is relative to an interpretation".

Are you with me, then, natural numbers and arithmetic truths are
_just relativistic_ notions (abstractions)?

From: Nam Nguyen on 26 Mar 2010 13:36

Alan Smaill wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Daryl McCullough wrote:
>>
>>> Ultimately, the people on this newsgroup who object to standard
>>> mathematics are really objecting to the idea that there can be
>>> such a thing as a counter-intuitive result.
>> That's a grossly erroneous over-generalization. It's many of those who
>> defend standard mathematics who erroneously object to the counter-intuitive
>> _but real nature_ of mathematics: relativity of truth and provability.
>>
>> To date they can't cite any absolutely true formula, without the formula
>> being false in another similar context, and yet they'd believe such
>> "absolute" truth is intuitive.
>>
>>> The ultimate logic
>>> would be one in which it is impossible to prove any result that
>>> you couldn't already guess was true.
>> The ultimate logic is one which is relativistic.
>
> Is that an absolute truth, then?
>
> I know, it's an old ploy, but your position just begs the question.
>

No. It's relative to what we, mortal beings, are entitled to know and to
what existence realm we happen to be in.

A lone man is walking in a road that seems to stretch to nowhere. Is the
evening lonely, or is that just a lonely feeling in the evening?