PROOF INFINITY DOES NOT EXIST! Not even ONE type
in [Math]
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From: Transfer Principle on 7 Jul 2010 21:36 On Jul 6, 12:23 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > INFERENCE there is no oo Herc has joined the growing number of sci.math finitists. We already know that Han de Bruijn and R. Srinivasan are finitists as well. A few posters, including Archimedes Plutonium and WM, defy classification. AP believes that all numbers larger than 10^500 are infinite -- so technically speaking, he still believes in infinity, but his infinity is much less than standard infinity. (Note that both Herc and AP use Chapernowne's constant in their arguments regarding infinity.) WM, on the other hand, believes that some standard natural numbers don't really exist, although he has no upper bound on the largest natural number. It had been said that for that reason, one can't call WM an _ultra_finitist, although it's possible that WM is merely a finitist. In the theory ZF-Infinity, one can't prove that an infinite set exists (assuming consistency), and if we add an axiom such as ~Infinity, we can actually prove that no infinite set exists. (There's no need to point out yet again that ~Infinity asserts that there is no _inductive_ set as opposed to no _infinite_ set. In several other threads, a proof in ZF-Infinity+~Infinity that all sets are finite has been presented, and the proof uses Replacement Schema. So let's not travel down that road for the umpteenth time.) Srinivasan has also proposed another axiom, D=0, which implies that all sets are finite, as well as a logic, NAFL, which proves the same. I recommend that Herc read some of the other sci.math finitists to see what they have to say. (Of course, he already knows about WM.)
From: Vesa Monisto on 7 Jul 2010 21:43 "Aatu Koskensilta" <aatu.koskensilta(a)uta.fi> wrote in message news:87d3uzi1oh.fsf(a)dialatheia.truth.invalid... > curt(a)kcwc.com (Curt Welch) writes: > >> But math isn't about simple physical existence. It's about words and >> their definitions. It's a study of formal langauge and what can be >> _said_ using a formal language. > > No it's not. After that categorical negation you could, I suppose, easily _say_ your categorical affirmation! -- "It is much more!", is not fair enough. -- If Curt's "10 goto 10" doesn't _say_ anything, it can _show_ that mathematics is many formal "language games getting their meanings from their us(ag)e" (_said_ Witt). -- That 'stepping' can even _say_ AND _show_ (ontic commitments AND cosmic 'workings'): 10 A = 0 20 A = A+1 : Print A : Goto 20 30 End Infinite stepping is here actualized as a potentiality (= possibility) even not as an actuality (hinted Aristotle already). -- Looping without Exit cannot be actualized to a 'finity' but can be programmed to the End. These 'levels' are easely confused (like levels of elemental sets and derived sets like powersets (combinations) by using same numeric codes. > Aatu Koskensilta (aatu.koskensilta(a)uta.fi) > > "Wovon man nicht sprechan kann, dar�ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus "Whereof one cannot speak, thereof one must remain silent", _said_ Witt in the last lines of Tractatus and wrote to Russell that it was "the main point" of the book, and "the cardinal problem of philosophy" (RUL 19.8.19). Oddly, Witt's meme might be seen as relevant in this thread: The saying / showing distinction might derive from Frege's paradox of concepts. Frege distinguished between objects or arguments, which are saturated, and concepts or functions, which are unsaturated (like 'infinity') i.e., cannot stand on their own but demand completion. In attributing properties to a concept we have to use a name to refer to something which is unsaturated, although names can only refer to saturated entities. The idea is that concept-words _name_ unsaturated entities but proper names cannot perform that role because they do not reflect the unsaturated nature of what they try to refer to. That is linguistic impotence, since the attempt to refer to concepts through names (e.g. 'The infinity') is a mistake forced on us by language (use, usage). Witt extends in Tractatus Frege's point by holding that _all_ names including those of objects, are unsaturated. -- Frege was worried about referring to unsaturated entities. Witt was worried about predicating of a symbol that it belongs to a logico-syntactic category. This worry arose out of reflections on Russell's theory of types. (RUL 1.13, NL 96...) Programmers might smile (at least) to the notions of "The infinity" and "ad infinitum", but it is possible that every electron is but a vortex stepping (looping) for to find "The Infinity" (= Exit to The Finity) but is stagnated to that never-ending cosmic commitment (= The Existence of lifetime oo). The proverb 'to exist' weren't but a bandpass-filter for cosmic commitments. As Barnsley wrote a book "Fractals Everywhere", so some physicist might write a book "Infinities looping Everywhere". -- What's that but 'existence'. V.M.
From: Curt Welch on 7 Jul 2010 23:07 "|-|ercules" <radgray123(a)yahoo.com> wrote: > "Curt Welch" <curt(a)kcwc.com> wrote ... > > Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: > >> "|-|ercules" <radgray123(a)yahoo.com> writes: > >> > >> > A counter example? > >> > >> Take the Heine-Borel theorem. It is not in any apparent sense "about > >> words and their definitions" nor can it be with any accuracy described > >> as "study of formal language and what can be said using formal > >> language". The study of formal languages and their expressiveness is a > >> very marginal part of mathematics. > > > > Well, the term "formal language" might be something very specific to > > you. I was using it very loosely just to mean a langauge made up of > > generally absolute concepts and definitions, instead of our normal > > natural language > > Your usage is fine. As long as all statements in the system can be > enumerated by some grammar. The fact you include 'study of words' makes > your definition of mathematics very apt. That was the problem with > systems like Godel's proof which was purely numerical and didn't consider > WHO verifies the proof. > > Aatu still believes formal systems will always be incomplete because > "computer123 cannot prove this statement" is true, but not provable by > computer123! > > Even though Aatu cannot verify yes or no whether "Aatu cannot prove this > statement" is provably true. I've not studied Godel's proof enough to feel like I really understand it's ramifications. But my best guess is that it's not really very significant at all. I think it just shows us an interesting fact of what can be done with language. When we make language self referential with a negative, we can create a form of logical "negative feed-back" that prevents the statement form being true or false. If we say "this statement is not true", we have created a negative self reference in our language. And once we do that, we have created a statement that can't be true, or false. The language is logically inconsistent with itself. I might be wrong, but I don't believe Godel's proof shows us anything more than the simple power of language to be self inconsistent when we write negative self referential statements. When we include language about whether something in the language is "provable" we make the problem more complex by looping the negative self reference through the external human (or machine) that is "doing the proof". But I believe the end result is no different than what happens when we write "this statement is not true". We are just using the same basic power of negative self reference in language to "mess up" the truth of any set of language statements. I think the only thing Godel shows us is that negative self reference is the poison apple of all formal language. > Herc > > PS > Isn't the fact you perceive something an absolute truth? Only if you believe your perception system works in absolutes - which there is no real evidence to suggest it does. Perception (and language) requires memory hardware of some form. How does the brain produce a statement such as "I see a rock" since the production of that language takes time which means there must be some for of "memory" of the detection of the sensory pattern of "rock" which persists long enough to produce that sequence of words. Are there any physical "memory" systems which can be said to perfectly flawless (to have perfect memory 100% of the time)? No, there is no such hardware becuase the very physical foundation of this universe is probabilistic in nature and not absolute in nature. Noise and randomness bubbles up from the bottom layers in the actions of all physical events. Computers attempt to be as absolute in their behavior as possible, but yet they are fighting a losing battle. They will always make an error at some point in time. The only question is how long, on average, can they run without making an error. The answer is that we build the machines so that number is larger enough, to make the computers practically very useful, even though in absolute terms, they are never perfect. All hardware is like that - nothing behaves in an absolutely predictable way 100% of the time. There is no indication that the human body is any different, which means our perception, and our memory of our perceptions, are never absolute. In fact, when carefully tested, we find human perception sucks. Our perception is full of errors, as well as our memory of our perception (if it's even valid to separate perception from memory but that's another question). If our perception and memory of perception is full of inconsistencies, then what would it mean to say that our perception is an absolute truth? If I _believe_ I saw a rock, what's the absolute truth in that if there was no rock? If then, a day later, I swear up and down that yesterday, I had said I saw a hat, not a rock, where's the absolute truth in the idea that I "thought" I saw a rock yesterday? And then, when shown the video tape of what I said yesterday, I come to believe I had said I saw a rock yesterday? But then, a day later, I find the video tape was unknown to me very carefully altered and in fact I had originally said I saw a hat? The point is, that our memory, and perception, is good enough, that most the time, we can trust it, and we do tend to treat it as if it were an absolute truth, but there's no real evidence to support the idea that there is anything absolute about it. We can't even trust that our memory remains constant for any period of time, so that we have no way of even talking absolutely about what our perception is. That is, we can't even trust that the words (or thoughts) we produce about our perception, is, or ever was, 100% consistent with what we perceived. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/
From: |-|ercules on 7 Jul 2010 23:26 "Curt Welch" <curt(a)kcwc.com> wrote... > "|-|ercules" <radgray123(a)yahoo.com> wrote: >> "Curt Welch" <curt(a)kcwc.com> wrote ... >> > Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: >> >> "|-|ercules" <radgray123(a)yahoo.com> writes: >> >> >> >> > A counter example? >> >> >> >> Take the Heine-Borel theorem. It is not in any apparent sense "about >> >> words and their definitions" nor can it be with any accuracy described >> >> as "study of formal language and what can be said using formal >> >> language". The study of formal languages and their expressiveness is a >> >> very marginal part of mathematics. >> > >> > Well, the term "formal language" might be something very specific to >> > you. I was using it very loosely just to mean a langauge made up of >> > generally absolute concepts and definitions, instead of our normal >> > natural language >> >> Your usage is fine. As long as all statements in the system can be >> enumerated by some grammar. The fact you include 'study of words' makes >> your definition of mathematics very apt. That was the problem with >> systems like Godel's proof which was purely numerical and didn't consider >> WHO verifies the proof. >> >> Aatu still believes formal systems will always be incomplete because >> "computer123 cannot prove this statement" is true, but not provable by >> computer123! >> >> Even though Aatu cannot verify yes or no whether "Aatu cannot prove this >> statement" is provably true. > > I've not studied Godel's proof enough to feel like I really understand it's > ramifications. But my best guess is that it's not really very significant > at all. I think it just shows us an interesting fact of what can be done > with language. When we make language self referential with a negative, we > can create a form of logical "negative feed-back" that prevents the > statement form being true or false. If we say "this statement is not > true", we have created a negative self reference in our language. And once > we do that, we have created a statement that can't be true, or false. The > language is logically inconsistent with itself. > > I might be wrong, but I don't believe Godel's proof shows us anything more > than the simple power of language to be self inconsistent when we write > negative self referential statements. > > When we include language about whether something in the language is > "provable" we make the problem more complex by looping the negative self > reference through the external human (or machine) that is "doing the > proof". But I believe the end result is no different than what happens > when we write "this statement is not true". We are just using the same > basic power of negative self reference in language to "mess up" the truth > of any set of language statements. > > I think the only thing Godel shows us is that negative self reference is > the poison apple of all formal language. There's half a dozen disciplines of maths based on negative self-reference. Step 1: define some collective set Step 2: negative self reference Step 3: claim SUPER elements exist outside the set Maths is actually really interesting if you believe all the nonsense! > >> Herc >> >> PS >> Isn't the fact you perceive something an absolute truth? > > Only if you believe your perception system works in absolutes - which there > is no real evidence to suggest it does. Perception (and language) requires > memory hardware of some form. How does the brain produce a statement such > as "I see a rock" since the production of that language takes time which > means there must be some for of "memory" of the detection of the sensory > pattern of "rock" which persists long enough to produce that sequence of > words. Are there any physical "memory" systems which can be said to > perfectly flawless (to have perfect memory 100% of the time)? No, there is > no such hardware becuase the very physical foundation of this universe is > probabilistic in nature and not absolute in nature. Noise and randomness > bubbles up from the bottom layers in the actions of all physical events. > Computers attempt to be as absolute in their behavior as possible, but yet > they are fighting a losing battle. They will always make an error at some > point in time. The only question is how long, on average, can they run > without making an error. The answer is that we build the machines so that > number is larger enough, to make the computers practically very useful, > even though in absolute terms, they are never perfect. All hardware is > like that - nothing behaves in an absolutely predictable way 100% of the > time. There is no indication that the human body is any different, which > means our perception, and our memory of our perceptions, are never > absolute. In fact, when carefully tested, we find human perception sucks. > Our perception is full of errors, as well as our memory of our perception > (if it's even valid to separate perception from memory but that's another > question). > > If our perception and memory of perception is full of inconsistencies, then > what would it mean to say that our perception is an absolute truth? If I > _believe_ I saw a rock, what's the absolute truth in that if there was no > rock? If then, a day later, I swear up and down that yesterday, I had said > I saw a hat, not a rock, where's the absolute truth in the idea that I > "thought" I saw a rock yesterday? And then, when shown the video tape of > what I said yesterday, I come to believe I had said I saw a rock yesterday? > But then, a day later, I find the video tape was unknown to me very > carefully altered and in fact I had originally said I saw a hat? > > The point is, that our memory, and perception, is good enough, that most > the time, we can trust it, and we do tend to treat it as if it were an > absolute truth, but there's no real evidence to support the idea that there > is anything absolute about it. We can't even trust that our memory remains > constant for any period of time, so that we have no way of even talking > absolutely about what our perception is. That is, we can't even trust that > the words (or thoughts) we produce about our perception, is, or ever was, > 100% consistent with what we perceived. I realize the difficulty in confirming a rock exists. But all you have to do is confirm *something* exists. Even if you're in error the conclusion is still true. E x Herc
From: Curt Welch on 8 Jul 2010 09:49
"|-|ercules" <radgray123(a)yahoo.com> wrote: > "Curt Welch" <curt(a)kcwc.com> wrote... > I realize the difficulty in confirming a rock exists. But all you have > to do is confirm *something* exists. Even if you're in error the > conclusion is still true. E x > > Herc That's an interesting point. I don't see any argument against the idea that something exists is an absolute truth. I think therefore I am. That might be the one and only absolute truth. But sadly, our ability to express the idea with language still runs into the problem of potential failure to correctly communicate with some small probability of error, and likewise, our ability to even think the idea comes with the same error. So even if the idea is itself, when expressed correctly, an absolute truth, it's not an absolute truth that our brain can ever correctly express or understand the idea. So maybe what we are left with, is that E x is only an absolute truth in the same sense that x = x is an absolute truth. That is, it's an absolute truth only by definition of how the language works. And since language, and more precisely the machine that interprets the language, can never function absolutely, the "idea" can't exist as an absolute truth. It only exists as a truth to the level of probability that the language processing machine can function correctly? I guess maybe in the end, what this means, is that relative to the language machine (us) that is processing the idea (having the though) it must always be an absolute truth because it can only have a truth value when the machine is functioning correctly, and at that point, the value must be true. But relative to a second party observing the first party, they will see that at times, the first party will fail to correctly process the language and result in the statement, and the idea being false. And then, what if there were points in time where the universe as we understand it stops existing? This might happen 1000 times a second every second with the universe we understand blinking in and out of existence constantly. But when we don't exist, we have no awareness of what has happened, and as such, our reality is just the reality of the times where we do exist, stitched together to create our view of the fabric of time. If our universe works that way, then what happens to the idea of E x? All human concepts of existence would be blinking on and off making E x false half the time. Clearly not an absolute truth in that sense. But to detect the fact that the human version of E x was not an absolute truth, there w2ould need to be a second (non human & non physical in our understanding of physical) langauge processing machine to notice, and think about the fact that the human understanding of E x was not an absolute truth. We currently have no way of knowing if our understanding of E x might suffer such a condition. As such, we have no way of proving that E x is an absolute truth for any given understanding of "existence" or "absolute". We can only define it to be an absolute truth we we limit our scope to within the framework of the language itself (which is saying we are limiting it to the times where the langauge processing hardware (our brains) are functioning correctly. -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/ |