Recent content by SSequence

I was looking at stevendaryl's informative posts related to this from few years back. It occurs to me that there is at least one way to think of associating a collections of sets [each set being subset of ##\mathbb{N}##] with ##PA##. One way is to look at all predicates of form ##P(x)## where...

I don't know anything about second-order, so I can't comment much on points related to that. However, it is worth mentioning that "second-order-arithmetic" can also often refer to a first order theory (often written as ##\mathrm{Z}_2##).

I don't know, for example, how closely the version of "untyped" lambda calculus [as, for example, described in notes linked in post#8] corresponds to the original one. Nevertheless, years ago I remember reading an interview. I don't remember it exactly, but this is what I re-call. Kleene...

I think I would add one point here. It seems useful to describe it since it has not been mentioned directly in the thread yet. Consider two circles ##C_1## and ##C_2## of radii ##r_1## and ##r_2## with their centers at points ##(x_1,y_1)## and ##(x_2,y_2)## respectively. Then the following...

Yeah you are right about that. This is the natural precise definition when one is using semi-formal prose as commonly used in math texts. Post#10 also mentioned this definition. I was just thinking about extra restriction of not using a sequence such as ##\{n_k\}## directly and then only relying...

I was thinking about the logical definition of sub-sequence using quantification on ##\mathbb{N}## alone. We are given sequences ##\{a_m\}## and ##\{b_n\}## and we want to determine whether the sequence ##\{b_n\}## is a sub-sequence of ##\{a_m\}## or not. Yesterday I was thinking about it and...

Not sure I should bump the thread for a small point. But I think I kind of get how the symbol ##:## seems pretty reasonable for use in logical statements (or representing predicates etc.). Though for longer expressions, personally I think it might be easier to use brackets (at least for me)...

I would write: ##\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\, [ x>y ] ## ##\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\, ( x>y ) ## It is also OK to write something like: ##\forall x \in \mathbb{Z} \, [\, \exists y \in \mathbb{Z} \,\, ( x>y ) ] ## ##\forall x \in...

Here are some of my thoughts on this topic. I have to say that I am not satisfied about my understanding of this (that is, looking at elementary equations from perspective of logic). There are number of things that can be confusing for me in this. Honestly, this is one of the reasons (amongst...

I was thinking about expressing the formula ##Prime:\mathbb{N} \rightarrow \{0,1\}## using ##\forall## instead of ##\exists## (although post#16 already does it). Continuing from post#17, I will again assume quantification over ##\mathbb{N}## in this post. One way just seems to be using the...

I did some calculations based on relations that I mentioned in post#9. Let ##p## be the probability of the player serving winning a point. Then we get the following: prob. of P1 winning from state (40,0) = ##S(40,0) \times p## = ##p^4## prob. of P1 winning from state (40,15) = ##S(40,15) \times...

Have you read the post#8 above (by erland). Outside of exception (like a tiebreak for which I don't know the full rule off-hand), in normal games you just have one player serving through-out a game (note that "game" is just part of a "set" and sets form a "complete game"). So essentially there...

I worked out the first part of the question using a spreadsheet. As a check I entered various values of probabilities and the probabilities do seem to add up to 1. One other check was setting the probability of serve winning as 0.5 and see whether the probabilities for both players winning the...

Suppose we want to express the predicate ##Prime:\mathbb{N} \rightarrow \{0,1\}##. We are assuming the set ##\mathbb {N}## to be: ##\mathbb{N}=\{0,1,2,3,4,.....\}## Assuming that the domain of quantification is understood to be ##\mathbb{N}##, here is one way of expressing the formula...

In what follows consistency is assumed through-out (to avoid adding a qualification to every sentence). At the very first level the following can be thought of as an answer to the question...