Recent content by agapito

OK thanks for responding

From elementary calculus it is known that (lim x-->0) ((sin x)/x) = 1. Is this result equivalent to (lim x-->0) sin x = x ? If so, how is it proved? Many thanks for all guidance.

Thanks. Unfortunately I cannot read your reply due to format of algebraic expressions. Is there any way to correct this so they display in "normal" format? Yes, They are read as "$(1,f'(x))$ to $1$", for example. I'm reading them in my Windows computer. Is there some way around this? Thanks

Thanks. Unfortunately I cannot read your reply due to format of algebraic expressions. Is there any way to correct this so they display in "normal" format?

Thanks for your guidance, Dan.

Many thanks for your reply, Klaas. $\Delta x$ is any real number. However, $dx$ is not "any real number", and it is also not a "real number of some magnitude". Where did you get that? From several calculus books. The existence of a separate "infinitesimal number" system is acknowledged but...

Question about the differential in Calculus. Assume a function y = f(x) , differentiable everywhere. Now we have for some Δx Δy = f(x + Δx) - f(x) The differential of x, is defined as “dx”, can be any real number, and dx = Δx The differential of y, is defined by “dy” and dy = f’(x)...

I posted this same question some time back and it was ably answered by Evgeny and Bachrack. My apologies for this oversight, I should have checked before posting. Agapito

One of the Peano Axioms specifies Sa = Sb --> a = b where S is the successor function. How does one establish from the axioms that S is, in fact, a function, that is the converse a = b --> Sa = Sb? Probably a very simple matter, but I would appreciate any help in clarifying. Many thanks...

Many thanks Evgeny. My interest in this is having a feel for what would be needed by a machine to perform these proofs. Obviously in that case no amount of metalanguage verbiage would help. A machine would have no way of "knowing" whether a certain symbol represents a function or something...

Thanks for responding. All descriptions of Peano Arithmetic I have seen indicate only the converse: Sx = Sy ----> x=y Can you please give me a reference containing x=y ----> Sx=Sy As a Peano axiom? Many thanks again, am

In axioms containg S one invariably finds: Sx = Sy -----> x = y The converse, which characterizes S as a function: x = y ------> Sx = Sy Is never shown. Neither is it shown as an Axiom of FOL or formal Theory of Arithmetic. From the basic axioms and rules of FOL, how does one go about...

What is the proper treatment of results about a formal axiomatized theory which are obtained from outside the theory itself? For example, there are 9 results dealing with the "≤" relation for Robinson Arithmetic, some of which are established by using induction, which is not "native" to Q...

Greatly appreciate your help with this, agapito

My text if Smith's Godel book. We establish that G (Golbach's conjecture and a Π1 wff of Robinson's Arithmetic Q) is true if and only if Q cannot prove ¬G. That much is clear. But then it goes on to say: G is true if and only if G is consistent with Q. We know that Q is sound (and thus...