The End of an Era - That of The Natural Numbers
in [Math]
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From: Daryl McCullough on 9 Jun 2010 00:07 Nam Nguyen says... >Daryl McCullough wrote: >> If U happens to be empty, then it immediately follows that >> ~Ex Phi(x) is true for every formula Phi(x). > >"immediately" how? I just went through that. Ex Phi(x) is *false* for the empty domain. ~Ex Phi(x) is true whenever Ex Phi(x) is false. >Especially in light of set membership of >an empty set and of Tarski's concept of truth? It *follows* from Tarski's definition of truth that Ax Phi(x) is true for the empty domain. That's what I just explained to you. -- Daryl McCullough Ithaca, NY
From: Aatu Koskensilta on 9 Jun 2010 00:13 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Weren't talking about whether or not "mathematical logic" is a "branch > of mathematics" when (or before) I asked you this question? Yes. Apparently you concluded from my explanation that the usual meaning of "branch of mathematics" has nothing to do with formal theories that I hold that mathematical logic has nothing to do with formal theories. I must admit I find your thought processes unfathomable once again. > Oh. So you are talking about English, like "potato chip", "couch potato", > and not about mathematical logic or reasoning! Well, it was you who wondered about the meaning of various English expressions. Naturally answering or ruminating about such questions requires that we talk about 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 9 Jun 2010 00:55 Daryl McCullough wrote: > Nam Nguyen says... > >> Daryl McCullough wrote: > >>> If U happens to be empty, then it immediately follows that >>> ~Ex Phi(x) is true for every formula Phi(x). >> "immediately" how? > > I just went through that. Ex Phi(x) is *false* for the empty > domain. ~Ex Phi(x) is true whenever Ex Phi(x) is false. How do you define "being true" using set-membership of the unformalized empty set? Until you could successfully define so, your "~Ex Phi(x) is true whenever Ex Phi(x) is false" isn't a valid argument in the unformalized set-membership paradigm. > >> Especially in light of set membership of >> an empty set and of Tarski's concept of truth? > > It *follows* from Tarski's definition of truth that > Ax Phi(x) > is true for the empty domain. Tarski's concept of truths is based on "concreteness" in the realm of abstraction (say of unformalized set) which says that: Ax Phi(x) is true <-> (There are x's) and (all x's are Phi(x)) which is of the form: A <-> B and C so if B is false (because U is empty), so is A. For an analogy: The raindrops are cold <-> (It rains) and (The raindrops are cold) <-> (The set of raindrops is non empty) and (cold(raindrops)) you observe that it rains not, so "The raindrops are cold" is false, as a _factual_ statement. And that has nothing to do with a _different_ logic paradigm (e.g. binary boolean algebra) where ~F = T. > > That's what I just explained to you. > Wrong kind of explanation though.
From: Nam Nguyen on 9 Jun 2010 01:13 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Weren't talking about whether or not "mathematical logic" is a "branch >> of mathematics" when (or before) I asked you this question? > > Yes. Apparently you concluded from my explanation that the usual meaning > of "branch of mathematics" has nothing to do with formal theories that I > hold that mathematical logic has nothing to do with formal theories. Didn't you say "It has nothing to do with formal theories", per the below? > 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? Aatu responded: > > You can easily find this out for yourself. It has nothing to do with > formal theories. > I must admit I find your thought processes unfathomable once again. Actually, as exemplified above, it's the other way around! Not to mention you didn't even offer an explanation! Your style of arguing, conversation doesn't seem like a "straight talk" and is strange. > >> Oh. So you are talking about English, like "potato chip", "couch potato", >> and not about mathematical logic or reasoning! > > Well, it was you who wondered about the meaning of various English > expressions. Naturally answering or ruminating about such questions > requires that we talk about English. You don't seem to understand: you used a natural language phrase in a highly rigid mathematical (reasoning) context, and your phrase was obscured and you need to technically explain that phrase. Otherwise it would be as "clear" as "incognitum"!
From: Aatu Koskensilta on 9 Jun 2010 01:25
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Didn't you say "It has nothing to do with formal theories", per the > below? Yes, I said the usual definition of "branch of mathematics" has nothing to do with formal theories. > Not to mention you didn't even offer an explanation! You can easily find out for yourself what is commonly meant by a branch of mathematics. > You don't seem to understand: you used a natural language phrase in a > highly rigid mathematical (reasoning) context, and your phrase was > obscured and you need to technically explain that phrase. Just reflect on Shoenfield some more and all will be clear! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |