Re: Scott and George's Teaching Thread
- From: george <greeneg@xxxxxxxxxx>
- Date: Thu, 23 Aug 2007 06:43:26 -0700
On Aug 20, 4:08 pm, Scott <ToaTe...@xxxxxxxxx> wrote:
Okay, let's start with the basics. What kind of notation do you want
me to use
The important thing to remember here is that there are general
conventions. What I or anyone else may want is less relevant than
what is generally done (in the real world).
The important thing, locally, is that, as MoeBlee
already explained, the real world uses multiple fonts and faces.
We (by contrast) have to do something that will fit within the
keyboard/character-set constraints of this channel.
for the following:
Variables:
lower-case letters from the end of the alphabet (about t..z).
Functions:
This is a little premature; we are doing set theory, which will
have a SECOND definition of "function" as "a set of ordered pairs".
BEFORE that, the definition of a first-order LANGUAGE had
function-SYMBOLS that could have -arities of ANY natural number
(they take a list of arguments, and the length of the list is fixed
for each function-WITH-A-SYMBOL-BUILT-INTO-THE-LANGUAGE).
These also need to be lower-case letters.
Quantification:Ax[whatever] and Ex[whatever].
This is also a keyboard-concession; in real life, the A would be
upside down and the E would be backwards.
Sets:This doesn't even matter; we will have an axiomatic theory
for constructing sets, so sets are just going to be terms in the
language of
that theory. There will be some special sets that are usually denoted
with
capital letters, like N and O (the naturals and the empty set).
The axioms of the theory will require the existence of certain sets
that arguably ought to have functions naming the thing that must
exist. Many formulations of those axioms do in fact have such
names; the power-set axiom, for example, is often written as
defining p(x). LOWER-case p. It's a FUNCTION[-symbol].
The theory will also have a Union axiom whose result will need to
be written U(x) (upper-case U) to distinguish the symbol from
a smaller binary union operator or from u as a variable.
But again, because you were new, as I said the first day,
the questions were premature. You did not know to even ASK
how to represent
PREDICATES.
P(x,y,z) is usual and typical but Px is acceptable as an alias for
P(x). It generally improves readability to omit parentheses if all
your names really are one letter. Obviously, if you have longer
names then you will have to put the commas and parens back.
We can say more once you start writing first-order
sentences yourself.
.
- References:
- Scott and George's Teaching Thread
- From: Scott
- Re: Scott and George's Teaching Thread
- From: Scott
- Scott and George's Teaching Thread
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