842: Gromov's "most powerful language" and set theory

[Joe Shipman]

Perhaps you are biased towards giving HF a special importance because it was your initial choice....

— JS

Sent from my iPhone

> On Feb 8, 2020, at 3:40 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
> 
> From a casual glance at the Gromov video I did not get a decent sense
> of what kind of "language power" Gromov is referring to. Under various
> well known interpretations of "language power" it is well known that
> set theory is much stronger than alternatives (because of the large
> cardinals that have such easy clean clear formulations). Even without
> the large cardinals set theory is at least as strong as alternatives
> under these usual interpretations.
> 
> Perhaps Gromov is talking about some sort of informal notion of
> language power. However, the video does spend a lot of time debunking
> mathematical discussions based on informal notions - even referring to
> such things as nonsense.
> 
> With regard to the present proper domination of set theory as a
> foundation, whereby category theory is interpreted in set theory,
> there is another feature that I have discussed some years ago on FOM.
> That is, that ZFC and probably the large cardinals as well are
> canonically generated from finite set theory. And finite set theory,
> with its principal hard core model HF (hereditarily finite sets) is of
> clear special foundational importance. .
> 
> 2/22/06  https://cs.nyu.edu/pipermail/fom/2006-February/009997.html
> 
> 2/25/06  https://cs.nyu.edu/pipermail/fom/2006-February/010063.html
> 
> 4/10/08  https://cs.nyu.edu/pipermail/fom/2008-April/012800.html
> 
> 4/14/08  https://cs.nyu.edu/pipermail/fom/2008-April/012800.html
> 
> 1/26/09  https://cs.nyu.edu/pipermail/fom/2009-January/013343.html
> 
> #######################################
> 
> My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
> https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
> This is the 841th in a series of self contained numbered
> postings to FOM covering a wide range of topics in f.o.m. The list of
> previous numbered postings #1-799 can be found at
> http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
> 
> 800: Beyond Perfectly Natural/6  4/3/18  8:37PM
> 801: Big Foundational Issues/1  4/4/18  12:15AM
> 802: Systematic f.o.m./1  4/4/18  1:06AM
> 803: Perfectly Natural/7  4/11/18  1:02AM
> 804: Beyond Perfectly Natural/8  4/12/18  11:23PM
> 805: Beyond Perfectly Natural/9  4/20/18  10:47PM
> 806: Beyond Perfectly Natural/10  4/22/18  9:06PM
> 807: Beyond Perfectly Natural/11  4/29/18  9:19PM
> 808: Big Foundational Issues/2  5/1/18  12:24AM
> 809: Goedel's Second Reworked/1  5/20/18  3:47PM
> 810: Goedel's Second Reworked/2  5/23/18  10:59AM
> 811: Big Foundational Issues/3  5/23/18  10:06PM
> 812: Goedel's Second Reworked/3  5/24/18  9:57AM
> 813: Beyond Perfectly Natural/12  05/29/18  6:22AM
> 814: Beyond Perfectly Natural/13  6/3/18  2:05PM
> 815: Beyond Perfectly Natural/14  6/5/18  9:41PM
> 816: Beyond Perfectly Natural/15  6/8/18  1:20AM
> 817: Beyond Perfectly Natural/16  Jun 13 01:08:40
> 818: Beyond Perfectly Natural/17  6/13/18  4:16PM
> 819: Sugared ZFC Formalization/1  6/13/18  6:42PM
> 820: Sugared ZFC Formalization/2  6/14/18  6:45PM
> 821: Beyond Perfectly Natural/18  6/17/18  1:11AM
> 822: Tangible Incompleteness/1  7/14/18  10:56PM
> 823: Tangible Incompleteness/2  7/17/18  10:54PM
> 824: Tangible Incompleteness/3  7/18/18  11:13PM
> 825: Tangible Incompleteness/4  7/20/18  12:37AM
> 826: Tangible Incompleteness/5  7/26/18  11:37PM
> 827: Tangible Incompleteness Restarted/1  9/23/19  11:19PM
> 828: Tangible Incompleteness Restarted/2  9/23/19  11:19PM
> 829: Tangible Incompleteness Restarted/3  9/23/19  11:20PM
> 830: Tangible Incompleteness Restarted/4  9/26/19  1:17 PM
> 831: Tangible Incompleteness Restarted/5  9/29/19  2:54AM
> 832: Tangible Incompleteness Restarted/6  10/2/19  1:15PM
> 833: Tangible Incompleteness Restarted/7  10/5/19  2:34PM
> 834: Tangible Incompleteness Restarted/8  10/10/19  5:02PM
> 835: Tangible Incompleteness Restarted/9  10/13/19  4:50AM
> 836: Tangible Incompleteness Restarted/10  10/14/19  12:34PM
> 837: Tangible Incompleteness Restarted/11 10/18/20  02:58AM
> 838: New Tangible Incompleteness/1 1/11/20 1:04PM
> 839: New Tangible Incompleteness/2 1/13/20 1:10 PM
> 840: New Tangible Incompleteness/3 1/14/20 4:50PM
> 841: New Tangible Incompleteness/4 1/15/20 1:58PM
> 
> Harvey Friedman