- Pink Floyd was/is the greatest band ever.
- Snowboarding is the most fun winter sport.
- Chess is very cool.
- I believe mathematics is quasi-empirical.
- Joe Satriani is the most awesome guitarist.
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10 comments
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December 1, 2007 at 4:15 pm
jayshree
hey would be nice if you’d teach me a bit of math. I like the idea of math.
December 20, 2007 at 3:08 pm
Paresh
Please list ZFC axioms using formulas only. Interpretations and comments are always confusing and they differ from from on instructor to another.
December 20, 2007 at 4:59 pm
topologicalmusings
Dear Paresh,
Thank you very much for your comment. I greatly appreciate it!
I had mentioned earlier in one of my posts, among other things, I had two purposes in mind before writing this particular series of posts on “Naive Set Theory (by Halmos).” First, to improve my own understanding of set theory (albeit from a naive point of view) and internalize the “knowledge” gained, and, second, to improve my expository skills in writing. I am not sure how far I have succeeded at exposition, but I think I have fulfilled the first purpose to a great extent.
In my view, for what it’s worth, simply listing the ZF (or ZFC) axioms doesn’t provide any information (read motivation) on how those axioms came to be created in the first place. I think it is really important to “create” mathematics by oneself instead of just reading and absorbing the “end products”, viz. the axioms. This naive understanding can then later be used as a springboard to study the formalized ZF set theory in a more “serious” way. (Also, there are lots of places on the Internet where one can find a list of the ZF axioms, anyway. My maintaining another list of those axioms wouldn’t really help anyone at all.)
By way of illustration, I am not highly satisfied with Halmos’ treatment of the Peano arithmetic in his book. It gets somewhat technical after a certain point and it is hard to see why the Peano axioms are introduced at all in the first place. After all, there are several other equally valid ways of constructing the set of natural numbers. However, I find Terence Tao’s treatment of the Peano axioms for building the natural numbers in his book, Analysis I, highly accessible and instructive. There, again, Tao demonstrates a naive way of understanding mathematics and then using formalism to “solidify” that understanding. I find that approach very useful in studying mathematics, especially when someone like me relies on self-study a lot.
Having said all that, I will surely list the ZFC axioms in one of my posts some time soon.
Thanks once again!
January 5, 2008 at 6:03 am
John Armstrong
Just finding your site now. Great work so far. And please by all means continue interpreting and commenting.
Wikipedia is a great place for just listing the axioms, and if someone wants them they can easily find them there. This weblog is your space for you to develop your voice. As you speak your own interpretations out into the air, they’ll solidify for you more than if they just floated in your head, and other people might find something worthwhile in them as well!
January 5, 2008 at 6:26 pm
Vishal
Dear Dr Armstrong,
Thanks a ton for your comments!
It certainly helps students like me whenever we receive some “encouragement” from professional mathematicians!
(And, I have taken care of the anchor tag in your previous post.)
January 8, 2008 at 4:10 pm
John smith
February 18, 2008 at 5:36 pm
eldila
First off, I love your blog. However, there is one thing that I find really annoying. Your rss feed doesn’t contain the entire article. This makes it impossible to read your blog by an rss reader. I would really appreciate it if you changed it.
Thanks
February 19, 2008 at 2:07 am
Vishal
Hi eldila,
Thanks for visiting my blog and loving it too! I have now set the Syndication Feed to “full text” for each article. Also, I don’t use the “more” tag for my posts. So, I hope things should be fine now. Please do let me know if the articles are shown as full-text in your rss reader.
Thanks for your feedback.
February 19, 2008 at 7:19 pm
eldila
Thanks for changing it.
Cheers!
May 17, 2008 at 8:09 pm
chronoplanet
Hi,
I am writing an article on empty collection intersection; I read the french version of the Halmos “Naive set theory”; I can’t have free access to the english version; could you send me the paragraph in the chapter “Complements and powers” where Halmos explain why the intersection of an empty collection gives the universe of discourse?
thanks a lot in advance;
best wishes,
Laurent Dubois
laurent.dubois@ulb.ac.be