tag:blogger.com,1999:blog-447603865959500290.post1578738832105582122..comments2017-09-26T09:06:50.702-04:00Comments on The TOF Spot: A Possible Short StoryTheOFloinnhttp://www.blogger.com/profile/14756711106266484327noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-447603865959500290.post-43812910085906050962017-06-27T18:50:04.418-04:002017-06-27T18:50:04.418-04:00Sooooo.......how does it end?
Sooooo.......how does it end?<br />Joe Keenanhttps://www.blogger.com/profile/17386544690405429208noreply@blogger.comtag:blogger.com,1999:blog-447603865959500290.post-18567230673043431022017-06-23T09:14:40.254-04:002017-06-23T09:14:40.254-04:00The O'Floinn,
If this is off-topic, I'm s...The O'Floinn,<br /><br />If this is off-topic, I'm sorry, but I had to ask this as there seemed to be no other venue to approach this with:<br /><br />What is the Thomistic interpretation of the advanced concepts of infinity in modern mathematics, such as Cantor's discovery that the infinity of the reals is bigger than the infinity of natural numbers? The latter being countable or listable or enumerable, the former being uncountable, unlistable and non-enumerable, meaning it is bigger?<br /><br />This discovery by Cantor implies there are different types of ininity, each biggger than the last. In fact, there is no biggest type of infinity, as you could just powerset the infinity that you have and end up with bigger and bigger ones, which also means that there is no such thing as a set of sets?<br /><br />Basically, the amount of infinities goes on for infinity, using standard mathematical notation.<br /><br />However, since standard mathematical notation does not give you a complete story, mathematicians have decided to jump beyond infinities that can be described using standard mathematical notation, and have reached cardinals bigger than any infinity you could formalise in a standard fashion.<br /><br />They are what is called ''large cardinals'', and there are various levels of those cardinals, which form an ascending hierarchy.<br /><br />Levels of large cardinals include innaccessible cardinals, indescribable, Mahlo, Vopenka, supercompact, huge, super huge, n-huge, super n-huge, all of which are bigger than the last.<br /><br />The biggest large cardinal mathematicians have discovered are the rank-into-rank axioms, while stronger cardinals are inconsistent with standard mathematical axioms and are thus rejected (Reinhardt, Berkeley cardinals).<br /><br />Now there is some controversy of whether or not the large cardinal hierarchy goes on forever, with some stating that they do not go on forever because of the danger of breaking with standard axioms, while others think that the cardinals go on forever without becoming inconsistent.<br /><br /><br />But either way, the large cardinal axioms (which do indeed make sense and are defended at length philosophically in the book ''Believing the Axioms'' ) are something that Thomistic mathematicians would have to adress<br /><br />And considering you are a mathematically inclined Thomist, what would be the interpretation of those higher level infinities, both Cantor's discovery of the listable and non-listable infinitie which go on forever via powersetting, and the large cardinals too,under the Aristotelian framework of mathematics that it uses?<br /><br />They cannot be merely abstract entities created by human reason, for Cantor's diagonal argument clearly shows that there is an objective difference in size between the infinity of natural numbers and the infinity of the reals, and the large cardinals follow a strict linear hierarchy that surprised mathematicians when it was discovered, which clearly implies objective properties that were not invented.<br /><br />Yet all of this talk of an infinite amount of infinities that can be arived at via powersetting, and the talk of cardinals bigger than even the infinity of infinities achieved via standard mathematical notation such as powersetting which are called large cardinals,are bound to seem strange to Thomists too.<br /><br />What are your thoughts on this as a Thomist who actually has lots of experience with mathematics and the concepts therein?JoeDnoreply@blogger.comtag:blogger.com,1999:blog-447603865959500290.post-13155090551217523382017-06-22T16:36:32.691-04:002017-06-22T16:36:32.691-04:00Funny, yesterday I was coinkadinkily mulling over ...Funny, yesterday I was coinkadinkily mulling over the idea of reading that alternative history again because a novice writer on Baen's Bar asked for a list of SF books to read.<br /><br /><br />JJBjjbrannonhttp://jjbrannon.livejournal.com/noreply@blogger.comtag:blogger.com,1999:blog-447603865959500290.post-42650582240990551372017-06-22T13:00:37.007-04:002017-06-22T13:00:37.007-04:00I'd read it. I liked these characters, Jacinta...I'd read it. I liked these characters, Jacinta in particular. Zachhttps://www.blogger.com/profile/09454555770949951135noreply@blogger.com