CUMC math presentation - need help!

[2218]

I have an undergrad math conference in Vancouver in July and I need to prepare a 25 or 50 minute speech to present at the conference. The audience is going to be mostly 3rd and 4th year students.

I want to present and develop the hyperreals using the compactness theorem, but I'm thinking this might be too much. I can't assume that these people have taken logic, so the compactness theorem requires an explanation of satisfiability, which in turn requires an explanation of models. If I brush over all that, then the construction will seem like I'm just pulling [censored] out of my ass.

Is this presentation doable? Is there some approach you guys could suggest? Or should I just pick a simpler topic?

[thylacine]

What are the hyperreals?
FWIW I understand the rest of that paragraph.

[2218]

[ QUOTE ]
What are the hyperreals?
FWIW I understand the rest of that paragraph.

[/ QUOTE ]

A field containing R as a subset (but no non-real complex numbers). It also contains infinite numbers (i.e. E b A r s.t. r is real and b>r, where E and A are the quantifiers) and their inverses, the infinitesimals.

[blah_blah]

fwiw i am one of the organizers of said conference

i don't know much about logic but it's not really a big deal to give a talk on something very advanced. I think that you should try to avoid getting too wrapped up in details though; 25 minutes is enough to give heuristic proofs and talk about the importance/implications of various results, but rigorous proofs eat up a lot of time that you may or may not have.

[2218]

[ QUOTE ]
fwiw i am one of the organizers of said conference

i don't know much about logic but it's not really a big deal to give a talk on something very advanced. I think that you should try to avoid getting too wrapped up in details though; 25 minutes is enough to give heuristic proofs and talk about the importance/implications of various results, but rigorous proofs eat up a lot of time that you may or may not have.

[/ QUOTE ]

I didn't plan on giving rigorours proofs. I learned this stuff two months into a logic class, and even then it was a lot to swallow. So to just jump straight into it seems like it will be a challenge.

I really wanted to show the compactness theorem because it is a beautiful one, and its use here is just ingenious. I doubt that is practical, however, for reasons mentioned in my OP.

I'm thinking of just mentioning the compactness theorem (not even stating it) as the underlying foundation and then saying something like: "'everything' true in R is true in *R (hyperreals) with the added fact that *R has an element, b, simultaneously satisfying the following (infinite set of) sentences {x>1, x>2, x>3, ...}," and then proceding with the construction.

Thoughts?

[PairTheBoard]

I think you need to decide whether you want your talk to be about Symbolic Logic and Model Theory or about the HyperReals. I don't think you will have time to give a full treatment of both. I don't think you can assume people know much about Symbolic Logic and Model Theory. In two stints of Graduate Studies in mathematics, 3 and 4 years, I was never exposed to the subject. Once you get the HyperReals you should be able to show some interesting things you can do with them. Do they really make things easier?

PairTheBoard

[Siegmund]

If your intention is just to show an interesting extension of the reals, I think you might find the surreals easier to develop without a bunch of background material than the hyperreals. It sounds to me from your first couple posts, however, that you're more interested in some of the background material you found of interest than in what you can do with the hyperreals.

Addressing your later question:
[ QUOTE ]

I'm thinking of just mentioning the compactness theorem (not even stating it) as the underlying foundation and then saying something like: "'everything' true in R is true in *R (hyperreals) with the added fact that *R has an element, b, simultaneously satisfying the following (infinite set of) sentences {x>1, x>2, x>3, ...}," and then proceding with the construction.

[/ QUOTE ]

...my first reaction to this is that you are going to have to spend at least a few minutes saying what you mean by "everything" (can you assume everyone has had abstract algebra and you can just throw around words like ring and field? lot of people didn't take abstract until after they took advanced calculus, where I went.) My second reaction is that anybody who doesn't know, or was taught but didn't fully grasp, the compactness theorem is going to look askance at the rest of your talk (or walk out, because they either can't follow it or don't trust it) because of the glossing over.

More generally, I have to comment that it's a real shame nonstandard analysis is talked about so little in undergraduate (and graduate!) classes. The cure for that is to make it intuitive and accessible, NOT to bury it under three years worth of theory before you start.

[Wyman]

At a UG math conference (or any conference for that matter), the audience does not expect to understand every detail of your talk. The most important things to do are

1) be excited about your material; your audience won't be if you aren't either.

2) present _something_ new that they can grasp. At the end of the day, they should know what the hyperreals are (at least generally, if not precisely) and why they are useful/cool.

3) do not get bogged down in technical details. If you present a proof, it should be short, cute, and comprehensible to a non-expert. In general, outlining a proof is much better.

IMO, of course.