The End of an Era - That of The Natural Numbers
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From: Aatu Koskensilta on 8 Jun 2010 23:10 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > In any rate, can you share with us what you think the "public" > definition of "branch of mathematics" be? You can easily find this out for yourself. It has nothing to do with formal theories. > But what are the naturals? 0, 1, 2, 3, and so on. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 8 Jun 2010 23:12 Daryl McCullough wrote: > Nam Nguyen says... >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Daryl McCullough wrote: >>>> >>>>> Because (Ax x=x) is true in every model. >>>> You meant every model where U is non-empty obviously. >>> No he didn't. >>> >> Well then he has yet to demonstrate the formula is true >> in a false model (where U is empty). > > I've explained it to you several times already. > > First, there is no such thing as a "false model". But the > way that truth in a model works for classical logic is this: > > If S is a structure for a language L, and Phi(x) is > a formula of L, then: > > If there is any way to assign a value to variable > x so as to make Phi(x) true, then the formula > > Ex Phi(x) > > is true in the structure. Otherwise, > > ~Ex Phi(x) > > is true in the structure. > > If U happens to be empty, then it immediately follows that > ~Ex Phi(x) is true for every formula Phi(x). "immediately" how? Especially in light of set membership of an empty set and of Tarski's concept of truth? > And also, > ~Ex ~Phi(x) is true for every formula Phi(x). > > Classically, ~Ex ~Phi(x) is interpreted to mean the same > thing as Ax Phi(x). So it follows that: > > if S is a structure for L with an empty domain, and Phi(x) > is a formula of L, then Ax Phi(x) is true in that structure, > and Ex Phi(x) is false in that structure. > > A special case is the formula x=x. So we have: > > If S is a structure with empty domain, then Ax x=x is true > in that structure. > > -- > Daryl McCullough > Ithaca, NY >
From: Nam Nguyen on 8 Jun 2010 23:27 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> In any rate, can you share with us what you think the "public" >> definition of "branch of mathematics" be? > > You can easily find this out for yourself. It has nothing to do with > formal theories. Well then you and I don't talk about the same thing. When I said "mathematical logic" I mean FOL which has a lot to do with formal theories. No wonder there's no communication here. > >> But what are the naturals? > > 0, 1, 2, 3, and so on. > Do you mean after 3, the naturals would be in what AP called as "Incognitum" zone (if I remember the terminology correctly)? In the name of _precise_ mathematical reasoning in FOL, what is the meaning of "so on".
From: Aatu Koskensilta on 8 Jun 2010 23:47 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Well then you and I don't talk about the same thing. When I said > "mathematical logic" I mean FOL which has a lot to do with formal > theories. No wonder there's no communication here. You didn't ask about "mathematical logic". You asked me to explain the usual meaning of "branch of mathematics". > In the name of _precise_ mathematical reasoning in FOL, what is > the meaning of "so on". I'm afraid there's not much I can do about your apparent inability to comprehend simple English. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 8 Jun 2010 23:59
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Well then you and I don't talk about the same thing. When I said >> "mathematical logic" I mean FOL which has a lot to do with formal >> theories. No wonder there's no communication here. > > You didn't ask about "mathematical logic". You asked me to explain the > usual meaning of "branch of mathematics". Weren't talking about whether or not "mathematical logic" is a "branch of mathematics" when (or before) I asked you this question? > >> In the name of _precise_ mathematical reasoning in FOL, what is >> the meaning of "so on". > > I'm afraid there's not much I can do about your apparent inability to > comprehend simple English. Oh. So you are talking about English, like "potato chip", "couch potato", and not about mathematical logic or reasoning! I see. |