Re: Extramathematical Notions and the Continuum Hypothesis

JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/9497
PC: http://permalink.gmane.org/gmane.science.philosophy.peirce/9498
JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/9499
KM: http://permalink.gmane.org/gmane.science.philosophy.peirce/9500
GF: http://permalink.gmane.org/gmane.science.philosophy.peirce/9505
JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/9512

Peircers,

Here are some notes from the beginning of my undergrad Prob & Stats textbook that I copied out
on a couple of previous occasions when this business about the probabilistic meaning of “event”
came up.  It appears that the other “o”-word I was trying to remember was “outcomes”.  I always
imagined that this use of “occurrences” and “outcomes” was the reason why probabilists typically
use a capital Omega for the name of the sample space.  At any rate, it makes for a good mnemonic.

Excerpts from 'Introduction to Probability Theory'
by Paul G. Hoel, Sidney C. Port, Charles J. Stone.

| 1.2.  Probability Spaces
|
| Our purpose in this section is to develop the formal
| mathematical structure, called a probability space,
| that forms the foundation for the mathematical
| treatment of random phenomena.
|
| Envision some real or imaginary experiment that we are trying to model.
| The first thing we must do is decide on the possible outcomes of the
| experiment.  It is not too serious if we admit more things into our
| consideration than can really occur, but we want to make sure that
| we do not exclude things that might occur.  Once we decide on the
| possible outcomes, we choose a set !W! [Omega] whose points !w!
| [omega] are associated with these outcomes.  From the strictly
| mathematical point of view, however, !W! is just an abstract
| set of points.
|
| We next take a nonempty collection $A$ of subsets of !W! that is
| to represent the collection of "events" to which we wish to assign
| probabilities.  By definition, now, an 'event' means a set A in $A$.
| The statement 'the event A occurs' means that the outcome of our
| experiment is represented by some point !w! in A.  Again, from
| the strictly mathematical point of view, $A$ is just a specified
| collection of subsets of the set !W!.  Only sets A in $A$, i.e.,
| events, will be assigned probabilities.  In our model in Example 1,
| $A$ consisted of all subsets of !W!.  In the general situation when
| !W! does not have a finite number of points, as in Example 2, it may
| not be possible to choose $A$ in this manner.
|
| Hoel, Port, Stone, 'Probability Theory', p. 6.
|
| Hoel, P.G., Port, S.C., & Stone, C.J.,
|'Introduction to Probability Theory',
| Houghton Mifflin, Boston, MA, 1971.

The notes continue here:

http://stderr.org/pipermail/inquiry/2003-June/thread.html#588

or here:

http://web.archive.org/web/20070301065551/http://suo.ieee.org/ontology/thrd13.html#04885

Regards,

Jon

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