Logic - 2017-10-15

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user131753

12:59

@user21820 I will be busy from 19th-23rd October. So I won't be able to participate. However, if you are able to do this any day before 19th and perhaps from 12pm (GMT) then I can participate.

user21820

@user170039 Hmm I am not free on weekdays. One alternative is that you can read the write-up I will post soon and then ask questions about it before or after the 'class'. Another alternative, if sufficiently many are interested and cannot make it during the first session, is for me to hold another session the week after. But that will have to depend on whether I've the time, as it would be close to some deadline I have to meet.

user131753

@user21820 I will go for the first alternative. Please ping me after your write up.

user21820

@user170039 Okay sure. I intend to post it by today. Almost finished.

user131753

@user21820 Are you using $\LaTeX{}$?

user21820

@user170039 Not in my write-up, but there is nearly no mathematical symbols so there's little need for it.

shredalert

13:16

I found this really good book on set theory that uses mappings as the primitive operation, instead of the inclusion operation

it's called Sets for Mathematics by Lawvere. Highly recommend it.

user21820

@shredalert I've heard of it. It's like category theory. Though I agree that functions should be considered more primitive than collections, and hence category theory is preferable in that aspect versus set theory, they both suffer from the same philosophical issue that makes me dislike current foundations haha..

Namely, in ZFC you can't have a universal set. In category theory you can't have a category of all categories.

shredalert

@user21820 Yeah, I agree functions should be more primitive. Cantor actually thought more in terms of mappings, but Zermelo, allegedly, didn't like that point of view.

I also agree about the universe problem

user21820

@shredalert Interesting. If you'd like to hear about my formal system when I'm satisfied with it, let me know. I really wish to solve both of this issues in as intuitive a way as possible. Namely in my system I will have the universal type as well as functions as sort of primitive, among many other things.

shredalert

I think when using predicate logic one must explicitly state which universe one is operating in

@user21820 Am definitely interested in reading about it once you're done.

user21820

13:43

@shredalert: Amusing; the syntax for bold text works only in the side-bar for pinned messages, and not where I want it to work, when the messages is a multiline one.......

Sigh, I will have to put the first copy in the trash.

→ 1 message moved to trash

shredalert

@user21820 that is quite an inconvenience

user21820

Yea, but it's okay. It's short enough and there's Ctrl+F.

Going to be away for quite a while.

See you later!

shredalert

see you later!

4 hours later…

user21820

17:18

=== Godel/Rosser's incompleteness theorem and simple computability proofs ===

Here I shall present very simple computability-based proofs of Godel/Rosser's incompleteness theorem, which require only basic knowledge about programs. Both proofs are by contradiction, and hence do not give an explicit independent (neither provable nor disprovable) sentence. In contrast, there are proofs that give an explicit independent sentence by using essentially a fixed-point combinator applied to provability, but that are quite a bit harder to understand than the unsolvability of the halting problem. It s…

→ 1 message moved to trash

Feeds

user21820 has added an event to this room's schedule.

user21820

I will hold a discussion type of class on sunday UTC 11am−1pm on Godel/Rosser's incompleteness theorem and simple computability proofs. All are welcome to join, but please read through the pinned post before attending!

If you can't attend, you can still post any questions here before the class, or read through the transcript of the class and then post any further questions, and I will answer them if I can. =)

shredalert

17:45

@user21820 I'll give it a read in the next few days.

user21820

Note that I don't expect readers to understand anything more than basic programming and basic logical deduction. So anything that isn't defined within the write-up itself is solely meant for interested people to explore further. =)

shredalert

@user21820 that's good to know actually

was hoping prerequisites wouldn't be too prohibitive

user21820

18:03

@shredalert If you find anything more than basic programming and logical deduction that seems crucial to understanding the theorems and their proofs (not necessarily all the side remarks) please let me know, and I can add an explanation. Same goes for if you can't quite see how some deduction is made.

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