@amWhy I'll give his stuff a look sometime. I've heard of 451 before.

@amWhy @shredalert: This post is senseless and needs delete votes if you've any to spare. Don't know why none of the other downvoters didn't vote to delete. Thanks! =)

And nearly all the other negative-score posts are crying for deletion as well...

@user21820 that's a lot of bad answers

@SimplyBeautifulArt: Yeap. If you can judge and help to delete them, all the better for Math SE!

Yeah...

@amWhy @SimplyBeautifulArt: Thanks for helping!

8-)

@user21820 so I had a conversation about infinity with a programmer the other day

Lol.

How did it go?

What kind of infinity by the way?

We were talking about the largest number, and he said infinity, so I said infinity plus one

Well, I introduced ordinals, and I think I really impacted a few people lol

@user21820 I don't understand what that question is about. Is the OP confused about what "if...then..." means?

And then I started talking about the epsilon numbers and they basically rebelled against me

@shredalert: Yes basically the asker thinks that "forall x in S ( P(x) )" only makes sense when S is non-empty.

Here S is the collection of people in front of you.

@SimplyBeautifulArt: I might too, if you go beyond countable stuff. =D =D

Heh, he said he didn't believe in infinity because it didn't work like a number

@user21820 Didn't anyone mention the truth table?

And that infinity plus one is still infinity

@user21820 in what way?

@shredalert Everyone used words to explain vacuous truth. And in some sense that's all one can do because there is no truth table that can explain quantifiers.

I can sorta take inaccessible ordinals in ordinal collapsing functions now

@user21820 I meant as in the truth table of implies.

@SimplyBeautifulArt I don't believe ZFC is meaningful, as I said before. I believe lots of systems are consistent and ZFC is likely to be one of them. But consistent systems can disagree with one another about arithmetical sentences.

I think a lot of people get confused about $A\Rightarrow B$ and think of it as "if A is true implies B is true" instead of "A implies B is true".

@shredalert Might as well get straight to the matter of the quantifier's meaning, rather than using analogy with implication, which isn't actually as clean as you think.

In particular, in non-classical logic implication may behave differently from restricted quantifiers.

@user21820 yeah...

I know it isn't clean, but I think a comparison with different types of if-then in natural language would benefit readers

But nonetheless, I think that inaccessible ordinals are too big for me at the moment

@shredalert: Well in this particular case, since implication wasn't brought up, I didn't see a need to. By the way, I don't understand why you say it is wrong to think "if A is true implies B is true".

It is in fact the correct way for both classical and intuitionistic logic.

@user21820 I'd rather we not participate or be ranked differently

Though you'd probably win

@SimplyBeautifulArt Oh my goodness. Of course I'm not going to participate, but even if I did I wouldn't win because I won't use a program I'm not convinced works. I wonder how such a contest would go anyway, since last Madore's attempted one on Reddit they just didn't bother to think carefully and gave nonsensical replies.

It was painful to read the replies just to get at the nice ones.

I want to raise the char cap to 512, so yeah, you'd probably go crazy

XD

And I don't think we're like Reddit

@user21820 I meant to say that most people think of A implies B as If A is true then B is true, and don't want to think of the other cases of truth values.

But that's the correct way to think of it.

It only makes a promise in one case; in other cases it says nothing at all.

This particular interpretation makes it compatible with intuitionistic logic.

Yes, but when A is False and B is False A implies B is still true

Not intuitionistically. =)

In this post I stick to classical logic, but you can separate out the game semantics from the law of excluded middle and see what I mean.

I'm talking about the formal version. Which beginners have trouble getting comfortable with. I did when I started out too.

You're talking about classical logic. And I agree we should just give the truth table to define classical implication.

But that's not the real meaning of implication.

That's why there is confusion in the first place. People do not realize how LEM plays a role.

There's a word for the type of meaning if-then we use in common speech

just a sec, looking it up in my book

Subjunctive conditional, that's the one.

Nah.. That's not true.

There are multiple English conditionals of the form "if... then..."

The subjunctive one is for possible situations.

There is an indicative one too.

Which is more or less the classical implication.

You're right, it's one of the possible ones, but I'm just saying it's a common one people think of when they see if-then

Hmm.. I'll have to think about whether people think of that by default, but you may be right.

I think the classical one in logic is called the material conditional

Yes that's the common name in philosophy, although I find it a ridiculous name.

I can't introspect because I automatically classify everything I say.

=P

haha

And I tend to use factual conditionals almost all the time.

And by the way subjunctive conditionals can be represented using modal logic.

Haven't got to that section yet

but it's in my book! :D

Though in general English sentences are difficult to translate to any formal logic due to immense context-dependency.

For example "There are people living in the US." looks so innocuous but "are people" is plural and "the US" implies uniqueness..

I bought Kleene's book "Mathematical Logic" today, because it has stuff on Church's type stuff and Godel, and Turing. My book doesn't cover those.

It was also very cheap - Dover.

I think I read some of Kleene's writings before. Clear.

Whoa! There is a button that lets me search through the chat!

At least on mobile web

Oh my you are the owner and very frequent user and didn't know?

Nope

Lol. Now you know how last time I found so quickly what you said earlier.

Lol, needed it so many times...

Yea!

I die lol

Yeah, the search on PC is a little bar in the top-right of the chat page.

:|

even says "search" :P

...

XD

I personally take Kleene's 3-valued logic as the correct logic to use for a system to capture natural language.

It avoids neatly all the paradoxes without having the problems with intuitionistic logic.

Is that what they call cube logic? Or some similar name

Searching for "cube logic" gives me Rubik's cube...

haha

Basically it's just true, false and undefined.

I vaguely remember reading about it somewhere

I'll look into fuzzy logic and topoi as well once I'm done with Henle's book.

Dover has a book on topoi and logic, so I think I'll just go for that when the time comes.

Fuzzy logic is another kind of logic, which is more applicable for systems that need to make decisions based on uncertain information.

Kleene's 3-valued logic is more abstract, for foundational issues.

I'm interested to see how fuzzy logic is put into practice.

Well it's easy to give you one example.. Just use probabilities as truth values.

Where 1 is True and 0 is False and everything in-between is like probability of being true.

yeah it's done with the interval [0,1] isn't it?

beat me to it haha

A and B = A*B ; not A = 1-A ; A or B = not ( not A and not B ).

And I think set membership works a similar way as well

This makes the assumption of independence, otherwise it's of course wrong.

indeed

Another possibility is A and B = min(A,B) ; A or B = max(A,B) ; not A = 1-A ; which is actually compatible with Kleene's 3-valued logic, taking undefined as 1/2.

People working with these kinds of logics have different ideas for how implication should be defined. It depends on the purpose, ultimately.

I was reading up on type theory and how it's used in programming languages, for example Haskell.

Yes.

I know a little bit of Haskell

But programming languages are based on very very weak type theories, usually so that type judgements (whether something has a certain type) are decidable.

The reason I went for Kleene is because his book covers most of the "core" stuff and is not a massive tome.

Haha..

He's concise.

Yeah, looks like it.

it had positive reviews from non-logicians as well, so that was another plus

As for sets, you can fix any set S and take subsets as truth values, and define True and False as S and {}, and define And,Or,Not as simply Intersect,Union,Complement. Then you get what is called a boolean algebra that satisfies classical logic!

I did all of that as an exercise in Courant's What Is Mathematics? A couple years ago :)

Ah nice.

That book has very good sections on calculus.

but pretty much nothing on algebra

It has a huge chapter on minimal surfaces, which I dip into every now and then.

The topics are elementary though.

I believe that was one of Richard Courant's favourite topics

I did a few soap-film experiments at home, it was a lot of fun. Made a big mess though. lol

@shredalert not as big as my numbers

@SimplyBeautifulArt No where near :P

@user21820 I might change the competition to include written mathematical stuff with same character limit so that non-coders can join

@shredalert I did soap-film experiments at some science exhibition or something. It's indeed interesting how the mechanical properties of the film lead to local surface area minimization.

And that it's not necessarily the global minimum.

@SimplyBeautifulArt I suggest you don't, because it would degenerate into what words are known concepts.

And if people on StackOverflow don't know how to code, they should learn how to code first before participating! =)

@user21820 true...

@user21820 :P

that was a good nap

Wherefore art thous? (thous is the plural form of thou).

reading :P

Making bumped ordinal collapsing functions

which I totally just made up

Do you have a Spring Break every year (might have already passed.

It's called Easter Break here.

I've been working on combinatorial outcome analysis with probabalistic implications.

Just been studying logic here, and doing assignments.

and reading a lot

@shredalert You didn't mention having fun, or relaxing....

reading fiction and playing games

there haha

or playing music

@shredalert Yay! for the fiction, playing games and/or playing music!

@SimplyBeautifulArt What about you? When is (was) your Spring Break?

@amWhy two weeks ago

@amWhy Do you understand my function above? If so, then can you help me figure out what BOCF(2) = ?

Oh, right, and $\Omega$ is the first uncountable ordinal

Off to bed. Goodnight all.

Good night @shredalert

Anything I can help you with @Zophikel ?

@Deedlit @user21820 Please look at this

I believe that BOCF(0) = e[0]

Hm, I think that BOCF(α) ≈ psi(α) until after we reach the supremum of the psi function

since B(BOCF(0),Ω) = e[Ω+1]

so we may continue to very high and scary places

@Simply Interesting I wonder how I'm even going to start to analyze your PAIN function

@Zophikel Analyze? As in try and grasp its magnitude? You learn about the fast growing hierarchy first

@Simply link

then you come back and look at the nightmare above

@Zophikel medium.com/@joshkerr/…

It has some typos and mess-ups, but it is a good introduction

@Simply all right

@Simply to you have a mathb.in with necessary defintions of your function

@Zophikel youtube.com/…

No :-(

PAIN is still a work in progress

and the above bumper function is still being looked at

The above YouTube channel is a few hours long, but worth it in my opinion

though possibly a little slow at first, it will blow your mind apart

@Simply dang current research

@Zophikel mathb.in/137162

Highly complicated

we recommend you watch the YouTube videos though, they are good

@Simply i'll have to watch them before diving into PAIN :)

Yes, you should

and after you dive into PAIN, you'll get a real feel of how incredibly large of a number you can make