tag:blogger.com,1999:blog-680838326158982647.comments2016-11-07T18:19:41.866-08:00What you say is true, SocratesWilliam Bellnoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-680838326158982647.post-27705020919382123092016-11-07T18:19:41.866-08:002016-11-07T18:19:41.866-08:00What is a number? Well supposing we just wanted t...What is a number? Well supposing we just wanted to talk about real numbers, then a number is a member of the set satisfying the following axioms:<br /><br />1. (R, +, ×) forms a field. In other words,<br />For all x, y, and z in R, x + (y + z) = (x + y) + z and x × (y × z) = (x × y) × z. (associativity of addition and multiplication)<br />For all x and y in R, x + y = y + x and x × y = y × x. (commutativity of addition and multiplication)<br />For all x, y, and z in R, x × (y + z) = (x × y) + (x × z). (distributivity of multiplication over addition)<br />For all x in R, x + 0 = x. (existence of additive identity)<br />0 is not equal to 1, and for all x in R, x × 1 = x. (existence of multiplicative identity)<br />For every x in R, there exists an element −x in R, such that x + (−x) = 0. (existence of additive inverses)<br />For every x ≠ 0 in R, there exists an element x−1 in R, such that x × x−1 = 1. (existence of multiplicative inverses)<br />2. (R, ≤) forms a totally ordered set. In other words,<br />For all x in R, x ≤ x. (reflexivity)<br />For all x and y in R, if x ≤ y and y ≤ x, then x = y. (antisymmetry)<br />For all x, y, and z in R, if x ≤ y and y ≤ z, then x ≤ z. (transitivity)<br />For all x and y in R, x ≤ y or y ≤ x. (totalness)<br />3. The field operations + and × on R are compatible with the order ≤. In other words,<br />For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)<br />For all x and y in R, if 0 ≤ x and 0 ≤ y, then 0 ≤ x × y (preservation of order under multiplication)<br />4. The order ≤ is complete in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words,<br />If A is a non-empty subset of R, and if A has an upper bound, then A has a least upper bound u, such that for every upper bound v of A, u ≤ v.<br /><br />Thought itself may be more universal than you're letting on.William Bellhttps://www.blogger.com/profile/02025125686289799066noreply@blogger.comtag:blogger.com,1999:blog-680838326158982647.post-79457599391438462712016-11-07T16:20:03.700-08:002016-11-07T16:20:03.700-08:00Philosophy is no more the most universal disciplin...Philosophy is no more the most universal discipline than thought itself. Even in our most basic truths we encounter no shortage of dilemmas. 1+1=2, yet what is a number? As definitions are proposed, the horizon is expanded. What then is definition? Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-680838326158982647.post-25296173527141604592016-06-03T08:55:30.823-07:002016-06-03T08:55:30.823-07:00Another:
the extreme of friendship is close to fr...Another:<br /><br />the extreme of friendship is close to friendship of oneself, there would seem to be friendship in so far as someone is regarded as a plural entity. (1166b trans Roger Crisp)William Bellhttps://www.blogger.com/profile/02025125686289799066noreply@blogger.com