Re: OT: Questions about Set Theory

Snit wrote:
Ok, Tim Adams and I have been discussing basic set theory... just curious
how knowledgeable others are about the topic. I have assumed this is
general knowledge but I may be wrong.

I'm knowledgeable about set theory. After a quick read of your post,
I don't see anything I disagree with. Terminologically, the words
"element" and "member" are more standard than "item." And generally,
people refer to "properties" of elements rather than "features."

Also, be careful about the distinction between a subset and an element.
For instance, the empty set is a subset of every set, but it's not an
element of every set.

Speaking informally about collections of things like you have is what's
called "naive set theory." Naive set theory is how most people, even
mathematicians, generally work. But occasionally someone serious about
this stuff learns a little of the technical foundation. It's much
the way a programmer most of the time deals with high level language
stuff, but occasionally thinks about lower level OS and machine
architecture issues.

The standard view is that set theory and logic together provide a
foundation for all of mathematics. Set theory is the language of
"things" and logic is the language of properties and reasoning.
These two interact in that logic allows us to reason about elements
of a set, and properties determine subsets of sets.

Mathematicians generally assume that any meaningful definition or
valid line of reasoning can be translated into something precise
in the languages of set theory and logic, though the translation can
result in something horrendously complicated. This is very analogous
to how a computer program written in a higher level language can be
compiled into machine code.

Also, this way of viewing the world seems quite likely to be hard
wired into the human mind. As you learn about set theory and logic,
you'll find that the basic concepts reflect all of the basic
constructions common to natural languages. (There's also a lot more,
or course.) Generally, nouns are sets. For instance, the word "man" can
be thought of as the set MAN of all men. "A man" means a member of the
set "man." Adjectives are properties or subsets of nouns: "white man"
specifies that the man has the property of being white, hence is in the
subset WHITE_MAN of MAN. Etc.


.

Relevant Pages

  • Re: OT: Questions about Set Theory
    ... I'm knowledgeable about set theory. ... the way a programmer most of the time deals with high level language ... "things" and logic is the language of properties and reasoning. ... basics of that? ...
    (comp.sys.mac.advocacy)
  • Re: Moore on Skolems Paradox
    ... their language, the proof of the uncountability of the reals is valid. ... mathematics is simply nothing whatever like ... If you want to imagine a world ... set theory would be true, but set theory says there are uncountably ...
    (sci.logic)
  • Re: Scott and Georges Teaching Thread
    ... The realm we are about to investigate, set theory, ... ONLY HAS ONE predicate. ... The definition of a first-order language gives rules for combining the ... The purpose of this enterprise is to prove theorems from axioms. ...
    (sci.logic)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... except that I meant my response to be in context of my ... theorems of category theory (if couched in the language of set theory) ... adequate for expressing ordinary mathematics. ...
    (sci.logic)
  • Re: Godel cant tell us what makes a mathematical statement true
    ... whatever actual name used), for example, both Enderton and Shoenfield. ... are not part of the underlying set language, ... language of set theory', then, yes, of course 'true' and 'false' are ... And to formalize Shoenfield's formulation set theoretically, ...
    (sci.logic)