what to do about the Successor axiom once it reaches 999....9999#318; Correcting Math
in [Logic]
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From: Nam Nguyen on 27 Jan 2010 01:49 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Let me clarify the definition: > > P(x) <-> Ey[y <= x] > > Infinite(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)] and of course: Finite(x) <-> ~Infinite(x) > >> If we take >> quantifiers to range over naturals there's a very simple definition of >> finiteness: >> >> x = x > > That's right I did think about this, in English: "All naturals are finite". > But let's not forget the context of defeating AP's crazy claim that his > definition (being a finite natural is being <= 10^500) is the best. Would > this be better than his? In a sense it's properly not: at least he had an > explanation that physically there seems to be some limit. If we force > him to accept "All naturals are finite", we'd have no way to explain > explain him why, based on the definition's own meriy, and he'd still > go on rambling nonsense. Imho In fact I myself once (in some thread) claimed something like "A natural number is neither finite nor infinite", simply because a natural number existentially isn't a set. So if I want to convince AP "All naturals are finite" I'd have no choice but come up with a *formal* definition, to contrast his crazy-non-formal definition.
From: Nam Nguyen on 27 Jan 2010 02:10 Nam Nguyen wrote: > Nam Nguyen wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Let me clarify the definition: >> >> P(x) <-> Ey[y <= x] >> >> Infinite(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)] > > and of course: Finite(x) <-> ~Infinite(x) > >> >>> If we take >>> quantifiers to range over naturals there's a very simple definition of >>> finiteness: >>> >>> x = x >> >> That's right I did think about this, in English: "All naturals are >> finite". >> But let's not forget the context of defeating AP's crazy claim that his >> definition (being a finite natural is being <= 10^500) is the best. Would >> this be better than his? In a sense it's properly not: at least he had an >> explanation that physically there seems to be some limit. If we force >> him to accept "All naturals are finite", we'd have no way to explain >> explain him why, based on the definition's own meriy, and he'd still >> go on rambling nonsense. Imho > > In fact I myself once (in some thread) claimed something like "A natural > number > is neither finite nor infinite", simply because a natural number > existentially > isn't a set. So if I want to convince AP "All naturals are finite" I'd have > no choice but come up with a *formal* definition, to contrast his > crazy-non-formal > definition. If we have L = L(<) then the set of all transcendentals would be a model of some T written in L, and in such case Ax[Infinite(x)] would make sense. But again no such thing as one shoe fitting all sizes.
From: Nam Nguyen on 29 Jan 2010 03:03 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Jesse F. Hughes wrote: >> >>> I think I underestimated your capacity for spurious self-defense. >> >> I had this definition: >> >> >>>>>>>> (*)P(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)] >> >> Clearly we can't talk about the truth or validity of (*)P(x) in, say, >> T = {(*)P(x) /\ ~(*)P(x)}. Also clearly is the _fact_ that (*)P(x) has >> a (typical) semantic/meaning. Could you tell me and others *just* the >> typical meaning of this formula (i.e. without mentioning anything about >> truth), and why '<' "does *not* stand for the usual less-than relationship"? > > I don't understand your question. The meaning is plain. And the reason you don't understand is because you got extremely confused between semantic and truth, and I'll explain that shortly. > > You wanted it to mean that x stands above infinitely many guys, but it > doesn't mean that. We'll see about that. (You should note however "mean" and "being true" are *not* the same!) > (Here, I say "x is above y" if y < x and I assume > you still intend y <= x to mean y < x or y = x.) > > P(x) is (Ey)(y <= x). Since x = x, this is trivially true. This is where you got very confused. "P(x) <-> (Ey)(y <= x)" is a plain definition, NOT a truh assertion, and we're supposed to talk about *semantic/meaning* of the expression. Didn't just say that I "wanted it to _mean_ that x stands above..." and that "The _meaning_ is plain"? Why then do you keep bring proof ("Since x = x") and model ("is trivially true"), despite my many times pleading with you not to? Do you not understand semantics and truth aren't the same? > > The remainder simply says that every y below x has another fellow > below him (and also x has a fellow below him). Thus, there are no > minimal elements below x. Isn't that what the concept of "infinity" is about, when expressed in L(<)? > > Obviously, this condition (that no element below x is minimal) is > unrelated to the finite/infinite distinction. Really? You stand on top of a ladder and there's *no minimal* rung below your feet, and the elevation of where you are isn't "infinite"? Would you call that "finite" height then? What do you think endless mean? What were you really thinking about when uttering "is _unrelated_ to the finite/infinite distinction"? Listen, Jesse, in: Infinite(x) <-> P(x) /\ AyEz[(y <= x) -> (z < y)] Semantically, I'd like to for Infinite(x) to _mean_: "x is a infinite [height] number" and conversely, ~Finite <-> ~Infinite(x) would mean: "x is a finite [height] number" Where must "is being true" be required in the definition? Is that so difficult to understand?
From: Nam Nguyen on 29 Jan 2010 11:56 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Isn't that what the concept of "infinity" is about, when expressed >> in L(<)? > > Certainly *not*. Every ordinal has a minimal element below it (namely > 0). I would not think that w ^ w is a finite ordinal. I'm clueless as to why you got so obsessed with models here ("ordinal"), when all of is is just a pure definition based on the semantic of non-logical symbols! An empty set, for example, is defined as: Empty(x) <-> Ay[~(y epsilon x)] where did you see the words "true", "model" in that definition? Let me turn the table around and ask *you to define* Finite(x), Infinite(x) than one couldn't find examples where it wouldn't make sense. (Read: there's no such thing as one definition fitting all contexts!!). And until you could, what you have been attacking is pointless and clueless.
From: Nam Nguyen on 29 Jan 2010 12:38
Nam Nguyen wrote: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Isn't that what the concept of "infinity" is about, when expressed >>> in L(<)? >> >> Certainly *not*. Every ordinal has a minimal element below it (namely >> 0). I would not think that w ^ w is a finite ordinal. > > I'm clueless as to why you got so obsessed with models here ("ordinal"), > when all of is is just a pure definition based on the semantic of > non-logical > symbols! > > An empty set, for example, is defined as: Empty(x) <-> Ay[~(y epsilon x)] > where did you see the words "true", "model" in that definition? Another example, meant to be about certain numbers, would be: Interesting(x) <-> (x=S0)-> (x+x=0)] NotInteresting(x) <-> ~Interesting(x) How would *you* decide the truth of Interesting(S0) in N and arithmetic modulo 2? Explain *why* you'd make such decisions? Do you now see the issue of defining concept based _solely on semantics_ of symbols? > > Let me turn the table around and ask *you to define* Finite(x), Infinite(x) > than one couldn't find examples where it wouldn't make sense. (Read: > there's > no such thing as one definition fitting all contexts!!). And until you > could, > what you have been attacking is pointless and clueless. |