details re Z-R |- PA consistent
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From: Transfer Principle on 15 Jul 2010 22:00 On Jul 14, 8:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Transfer Principle wrote: > > Here are my attempts to write Herc's axioms more rigorously: > > Attempt #1 (Schema): > > (phi(0) & (Ax (phi(x) -> phi(xu{x})))) -> phi(I) > "Rigorously"? So what is '0' in this set-language? and what are the > axioms concerning it? Here 0 refers to the von Neumann ordinal zero, also known as the empty set. We can eliminate this symbol as follows: Schema: Ax (((Ay (~yex)) -> phi(x)) & (phi(x) -> phi(xu{x}))) -> phi(I) But I is a primitive symbol, intended to represent the set of all naturals whose existence Herc accepts. Herc himself first came up with the symbol I. > > Attempt #2: > > {} _is_ a (Frege) natural number. > > (This axiom has been challenged by Jeffries, though.) > In term of rigorousness, this is even worse than your attempt #1 > above! With Jeffries, Nguyen, and MoeBlee all questioning this axiom, it's best that I drop it. But Attempt #1 is a schema, and to me it's awkward to use a schema (which is a hybrid of several schemata mentioned by either Herc or me) to describe the _finite_ set I. This is why I proposed Attempt #2, a single axiom, but that axiom depends on the existence of a universal set V (as in NFU) which is, admittedly, starting to drift away from Herc's original intention. So I drop Attempt #2. > Again, why don't you state your own axioms? Good question. As I often point out, it had never even occurred to me that there can be alternate theories until I saw "cranks." Indeed, the first alternate theory I saw was proposed by a "crank" on a website other than sci.math, a theory with combines TO-style set size with AP-real-style infinitesimals. Even if I drop the Herc and AP proposals due to their proponents' unorthodox (or to use a five-letter word, "crazy") non-mathematical ideas, the other ideas that I'm currently considering (either in active or future threads) all go back to ideas first proposed by other posters. These include the following proposals. (I give the desiderata, followed by the original proponents of the idea in parentheses.) 1) There exists a singleton tower -- a set that has no transitive closure (zuhair, discussed by Libert). 2) The natural numbers are defined as: 1 = {1} 2 = {1,2} 3 = {1,2,3} 4 = {1,2,3,4} etc. (several posters who prefer the naturals start with 1 rather than 0. Notice that ZFA doesn't work because it, assuming consistent, can't refute that these are all equal). 3) I had earlier dropped TST due to its reliance on three-valued logic, but I might reconsider it because of a possible patch to avoid three-valued logic (tommy1729, with the patch extracted from a Srinivasan post). All of these come from earlier posters, and so none of them are original to me. Indeed, the poster zuhair has come up with so many ideas for theories that anything that I might cook up is likely to have been previously mentioned in a zuhair thread. Then again, if I just threw together some random symbols (taking care to define and use the correct arities, etc.), the resulting axioms are likely to be original. But any axioms created for the sole purpose of being original are extremely likely to prove very many interesting theorems, or being even remotely applicable to the sciences. Therefore, I'm highly dependent on other posters to come up with ideas for axioms and theories.
From: Transfer Principle on 15 Jul 2010 22:22 On Jul 15, 7:40 am, James Burns <burns...(a)osu.edu> wrote: > Transfer Principle wrote: > > Instead, I want to consider those who shut out their > > opponents' ideas to be closed-minded. > One of the characteristics very commonly attributed those > called cranks is their resistance to ideas opposed to > their particular brand of crankishness. Do you consider > this characteristic evidence of closed-mindedness > (setting to one side for a moment whether those so called > exhibit this characteristic)? > If you do not think it shows closed-mindedness, then it > looks to me as though you show considerable closed-mindedness > of your own in how you assess others' closed-mindedness. Interesting question, and not an easy one for me to answer. In considering whether finitists such as Herc or WM are closed-minded, I often wonder whether _every_ finitist would be considered closed-minded. For infinitists typically accept the existence of finite sets, but finitists reject the existence of infinite sets. So in a way, it could be argued that finitism in itself is a closed-minded idea. Thus, my answer to Burns's question depends on whether open-minded finitism is possible. If not, then Burns is right that by not seeing this, I'm being closed-minded myself, and the remedy would me to stop defending finitists and defend only infinitists -- though I reserve the right to defend non-standard infinitists. If open-minded finitism is possible, then I can follow the suggestion given by Burns: > If you do think it shows closed-mindedness, then you have > been ignoring an important part of the task you have claimed > for yourself. You should be showing the wider audience of > sci.math and sci.logic that Herc, WM, whoever else > you want to add to the list are open to "standard" > ideas and are not clinging to their own out of mere > stubbornness. In doing so, I would have to know how to be an open-minded finitist, so that Herc and WM can be more open-minded in their finitism and I can be more open-minded in their defense.
From: Transfer Principle on 15 Jul 2010 23:19 On Jul 15, 9:59 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 14, 8:31 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Attempt #2: > > {} _is_ a (Frege) natural number. > For convenience, please define again "is a Frege natural number" in > the language of set theory (and if in some other formal language, then > please specify it). By this point, Jeffries and Nguyen have convinced me not to consider Attempt #2 anymore as it's based on NFU, and this might be too far removed from what Herc originally intended. Thus, MoeBlee can save himself time by avoiding the following explanation on what a Frege natural number is. Still, if anyone is still interested, here is how the set theorist Randall Holmes defines natural numbers: > Also, I take it that '{}' stands for the empty set. It does. Next, Holmes defines (the Frege natural) 0 to be {{}}. Next, he defines successor, as follows: "For any set of sets A, we define A+1 as the collection {au{x} | aeA & ~(xea)} of unions of elements a of A and singletons of objects x not in a." Next, he defines the successor ("increment") function, as follows: Inc = (A |-> A+1) (Notice his notation -- this means that Inc is the function which maps A to A+1. It's the set of all ordered pairs of the form (A,A+1).) Next, he defines the set of all inductive sets, as follows: "Ind, the collection of _inductive_ sets, is defined as {A | 0eA & Inc[A] subset A}; an inductive set is one which contains 0 and contains the 'successor' of each of its elements." (emphasis his. Also notice his notation -- Inc[A] is the image of the set A under the function Inc, so the elements of Inc[A] are the successors of the elements of A.) Next, he defines the set N in the expected manner: "N, the collection of natural numbers, is defined as Intersect[Ind], the collection of objects which belong to all inductive sets." Finally, we define the predicate "is a Frege natural number": n is a Frege natural number <-> neN To repeat, Holmes uses all these definitions in NFU. In ZFC, none of these objects (except {{}}) are sets. Holmes's definitions can be found at his website: http://math.boisestate.edu/~holmes/holmes/nf.html > > "New Axiom: define finite-Natural Number as all less than 10^500 (the > > largest meaningful > > number in physics) and define infinite Natural Number as all those > > equal to or larger than 10^500." > Please define 'meaningful number in physics' in the language of set > theory By its very nature, "meaningful number in physics" can't be defined within set theory, but outside of set theory -- namely, within _physics_. This theory does seek to axiomatize math for the sciences, and _physics_ is exactly the science that this theory seeks to axiomatize. Of course, AP could have simply written: "New Axiom: define finite-Natural Number as all less than 10^500 and define infinite Natural Number as all those equal to or larger than 10^500." but then the reader would be left wondering whether the number 10^500 was arbitrarily chosen. So AP adds "largest meaningful number in physics" to emphasize that the upper bound was not arbitrarily chosen, but we must go outside of set theory and into physics to find its significance.
From: Transfer Principle on 15 Jul 2010 23:56 On Jul 15, 11:45 am, James Burns <burns...(a)osu.edu> wrote: > Transfer Principle wrote: > > As for AP, he's recently provided the following axiom: > > "New Axiom: define finite-Natural Number as all less than > > 10^500 (the largest meaningful number in physics) and > > define infinite Natural Number as all those equal to or > > larger than 10^500." > It happens that the number of ways 8 poker decks of cards A suggestion: let's say "blackjack" instead of poker, since poker is only played with a single deck, while blackjack is regularly played with as many as eight decks. Thus, it can be said that Burns is describing a real-life scenario. > (with identical backs) can be shuffled together is one of > these infinite Natural Numbers, > (8*52)!/(8!)^52 =~ 1.25e671, > as one of us Standard Theorists might oppressively > put it. Do you or AP intend to distinguish that infinite > Natural Number from any other infinite Natural Number? > What if I were to destroy one card from the collection > of 8 decks and shuffle again? A Standard Theorist might > say that there are > (8*52-1)!/(7!*(8!)^51) =~ 2.41e669 > ways to shuffle the resulting deck. As far as you and AP > are concerned, though, this is just another infinite > Natural Number -- or so I understand you. When considering these questions that are finitistic or ultrafinitistic in flavor, we can consider Yessenin-Volpin, the famous ultrafinitist. I know that Y-V is respected here at sci.math much more than Herc or AP is. I will not ask _why_ Y-V is so respected, since I know that it will lead to another go-nowhere discussion. Instead, I will take _advantage_ of the fact that Y-V is respected here at sci.math, and so I'll answer Burns's question in terms of Y-V's ideas. To repeat, Burns asked: > Suppose I asked whether there were more, fewer, or > the same number of shuffles in the first instance > as in the second instance. Am I even allowed to ask > this question? My answer to Burns's question "Am I even allowed to ask this question" is that he is allowed to ask it if and only if Yessenin-Volpin would answer it, and its answer is the answer that Yessenin-Volpin would give. Thus, I have recast AP framework in terms of the Y-V framework, since Y-V is more likely to be accepted. To repeat, we know that when Y-V is asked the question "Is n a natural number?" he would wait until a time interval f(n) has passed before answering "yes." And so we can ask ourselves what f(1.25e671) is. We can give Y-V an extremely long time limit, say the lifespan of the universe, so that if f(1.25e671) is less than this lifespan, then Y-V would give the classical answers to Burns's questions. Burns's question, meanwhile, is typical of those asked of people like AP and Y-V. Whenever someone attempts to give an upper bound on the largest number that can appear in a real-life scenario, someone like Burns will give a real-life scenario (such as a blackjack game) in which a larger number appears. To this end, one could argue that Herc was _wise_ not to answer my question about the largest number in I, since as soon as Herc gives one, someone will jump on it and challenge the upper bound. But what if I were to give an upper bound that's much, much larger than the one given by AP? For example, what if I were to come up with a theory in which the famous Graham's number is the largest natural number? Burns would be hard-pressed to come up with a real-life situation which required a larger number than Graham's. Of course, he could always say something like, "the number of ways that Graham's number of blackjack decks can be shuffled is greater than Graham's number," but this is pushing it since one can't physically shuffle Graham's number of decks in the same way that he can shuffle eight decks. Thus, there really is an upper limit to the largest number that can appear in a physical situation. This post establishes that this number is between 1.25e671 and Graham's number. So as soon as we agree upon this upper bound, then we can consider an alternate theory which proves that number to be the largest finite natural number.
From: Jesse F. Hughes on 16 Jul 2010 09:22
Transfer Principle <lwalke3(a)lausd.net> writes: > A suggestion: let's say "blackjack" instead of poker, since poker > is only played with a single deck, while blackjack is regularly > played with as many as eight decks. Thus, it can be said that > Burns is describing a real-life scenario. The standard, 52 (54?) card deck is commonly called a poker deck. Besides, what would it matter whether a standard game uses the eight decks or not? It's still a perfectly reasonable scenario. A human being can, in a reasonable amount of time, shuffle eight decks together. -- Jesse F. Hughes "Well, I don't claim to be an expert, in fact I am a fry cook with a national burger chain, but I have solved many differential and partial differential equations numerically." --C. Bond |