Axiom of Determinacy - Infinite Logic and The Axiom of Determinacy

Infinite Logic and The Axiom of Determinacy

Many different versions of infinitary logic were proposed in the late 20th century. One reason that has been given for believing in the axiom of determinacy is that it can be written as follows (in a version of infinite logic):

OR

Note: Seq(S) is the set of all -sequences of S. The sentences here are infinitely long with a countably infinite list of quantifiers where the ellipses appear.

In an infinitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantifiers that are true for finite formulas, such as OR  \exists a: \forall b:
\exists c: \forall d: \lnot R(a,b,c,d).

Read more about this topic:  Axiom Of Determinacy

Famous quotes containing the words infinite, logic and/or axiom:

    Let the will embrace the highest ideals freely and with infinite strength, but let action first take hold of what lies closest.
    Franz Grillparzer (1791–1872)

    There is no morality by instinct.... There is no social salvation—in the end—without taking thought; without mastery of logic and application of logic to human experience.
    Katharine Fullerton Gerould (1879–1944)

    The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood.... For the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid.
    Edgar Allan Poe (1809–1845)