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ruveyn
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14 May 2012, 10:58 am

techstepgenr8tion wrote:
I don't think we're limiting ourselves.

To say that something is both true and false or neither true nor false (which I don't know what criteria would even separate those two), to me just says that the matter being discussed hasn't properly been digested and broken down into its individual factors - hence you have one thing that's true, one thing that's false, its really two things that you're mistaking for one thing.


As in true on one sense and false in another sense. We call that ambiguity. The curse of all fuzzy language usage is ambiguity. The only thing we can reasonably require is that an assertion not be a logical contradiction or lead to a contradiction by inference.

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b9
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14 May 2012, 11:06 am

Robdemanc wrote:
The reason I ask is because I saw a documentary about infinity. It stated that if we had two infinte sets, can we say they are both equal?

Example:

A = {The set of all numbers}

B = {The set of all even numbers}

Can we say that A > B? If both sets are infinite how can we say one is greater than the other? Perhaps we could say that the statement A > B is neither true nor false, or is both.


there are no limits on A or B, so they can never be quantified, and therefore can never be compared.



Robdemanc
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14 May 2012, 12:49 pm

Species5618 wrote:
Robdemanc wrote:
I am not asking here about what is good or bad. I am talking about our mindset that things have a true or false value.

The reason I ask is because I saw a documentary about infinity. It stated that if we had two infinte sets, can we say they are both equal?

Example:

A = {The set of all numbers}

B = {The set of all even numbers}

Can we say that A > B? If both sets are infinite how can we say one is greater than the other? Perhaps we could say that the statement A > B is neither true nor false, or is both.


The point here is that you do not define what ">" is. When you compare 2 numbers, the meaning of > is clear, the standard greater-than relation. For sets of arbitrary size, however, there are multiple ways to interpret >.

Some other posters in this thread have already mentioned comparing both sets on their "cardinality" (a technical term for size, which is extended for infinite sets). If you look at cardinality, then A = B. If you interpret the > sign as denoting "is a superset of", then A > B is true.

In mathematics, a statement is always either true or false. Not both or neither. But that does require that the statement is very detailed and that all terms are well defined. There is a small branch of math called "fuzzy logic", where a statement can also take on truth-values between the two extremes of true and false (if 0 is false and 1 is true, fuzzy logic allows the truthfulness of a statement to be any real in [0,1], allowing a statement to be "somewhat true"), but this is a sidetrack of mainstream math.

In real life, the question of binary truth is obscured by the fact that many statements are not factual and are polluted with subjective elements. "The weather is nice today" can be both true and false depending on who you ask.


Sorry

> means greater than. So was asking if set A is greater than set B.



ruveyn
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14 May 2012, 2:58 pm

Robdemanc wrote:

Sorry

> means greater than. So was asking if set A is greater than set B.


Still ambiguous. Does > mean a super-set or does it mean a set of greater cardinality.

A is a super set of B. A and B have equal cardinality.

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14 May 2012, 3:54 pm

You cannot correctly use operators created for finite numbers when dealing with transfinite numbers. Speaking of whether the set of natural numbers is "greater than" the set of even natural numbers is a meaningless exercise if is approached with the same analysis that one uses for finite sets.

The entire argument of this thread seems to me to be predicated on a question of whether (and when) truth or falsity can be absolute.

Mathematics is one of the few fields in which absolute truth is possible--because mathematics exists in an entirely artificial environment in which all concepts are strictly defined--either by axiom, or by proof and construction. Therefore, mathematics can be subject to absolute analytical rigour in which a hypothesis is true if proven true, and always true thereafter (subject, of course to change in the environment--absolute truth in plain geometry is not necessarily so in spherical geometry, for example).

But other discplines are not subject to the same type of rigour. Engineering and economics are replete with error terms, because the mathematical models that govern them are imprecise--mathematically perfect, but imprecise analogs of the real phenomena they purport to predict.


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techstepgenr8tion
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14 May 2012, 8:07 pm

ruveyn wrote:
techstepgenr8tion wrote:
I don't think we're limiting ourselves.

To say that something is both true and false or neither true nor false (which I don't know what criteria would even separate those two), to me just says that the matter being discussed hasn't properly been digested and broken down into its individual factors - hence you have one thing that's true, one thing that's false, its really two things that you're mistaking for one thing.


As in true on one sense and false in another sense. We call that ambiguity. The curse of all fuzzy language usage is ambiguity. The only thing we can reasonably require is that an assertion not be a logical contradiction or lead to a contradiction by inference.

ruveyn

The two, three, or more different 'things' could also be relativities. Something could be true when held in one situational context, false when held in another; doesn't make it true-false or neither true nor false, just means 'it depends'.


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ruveyn
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14 May 2012, 8:09 pm

techstepgenr8tion wrote:
The two, three, or more different 'things' could also be relativities. Something could be true when held in one situational context, false when held in another; doesn't make it true-false or neither true nor false, just means 'it depends'.


If context isn't everything it is a large part of everything.

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techstepgenr8tion
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14 May 2012, 8:19 pm

It's also the wild west of truth, utter falsehood, and the 99.9999 of every else in the way of half-baked notions where one thinks they're on solid ground one minute to find out that they're simply slipping and skating on another beltway meteorite with no up, down, back, or forward again. Science has made an incredibly valiant attempt at mapping and chaining as many meteors together but it's still just a substantial local piece that's slipping and sliding itself within the larger system.

Labeling orders of saliency/magnitude, causality, and effect right seems to be the biggest struggle in terms of unifying knowledge. Regardless of what we come up though I get the impression - so long as our rate of discovery doesn't stagnate, that's its quite possible that many views even held today in most directions will be somewhat flat-earth'ish by the end of the 21st century.


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14 May 2012, 8:27 pm

visagrunt wrote:
Mathematics is one of the few fields in which absolute truth is possible--because mathematics exists in an entirely artificial environment in which all concepts are strictly defined--either by axiom, or by proof and construction. Therefore, mathematics can be subject to absolute analytical rigour in which a hypothesis is true if proven true, and always true thereafter (subject, of course to change in the environment--absolute truth in plain geometry is not necessarily so in spherical geometry, for example).



Ponder the Goedel Incompleteness Theorems.

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14 May 2012, 9:54 pm

Pondering Gödel doesn't make visagrunt's post less valid. What do you mean, ponder?



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15 May 2012, 11:36 am

I am not sure that Gödel is inconsistent with my observation about the avaialbility of absolute truth in mathematics.

Gödel stands for the proposition that a formal system cannot be both complete and consistent. But I have never claimed that absolute truths in mathematics are complete--only that once proved, they are then absolute--within that same axiomatic system. But as soon as you change the axioms, then the truth of previously proved statements goes out the window.


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15 May 2012, 11:12 pm

Never mind.... what Declension and ruveyn said.



Last edited by heavenlyabyss on 15 May 2012, 11:20 pm, edited 1 time in total.

Declension
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15 May 2012, 11:17 pm

EDIT: No longer relevant.



Last edited by Declension on 15 May 2012, 11:36 pm, edited 1 time in total.

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15 May 2012, 11:18 pm

Yeah, I was just about to edit my post because I realize you answered the question already... what you are saying makes sense.