3 ms·

Orders on propositions are crucial for the consistency of foundations for reasons explained in the following article: "Epistemology Cyberattacks" https://pap

Orders on propositions are crucial for the consistency of foundations for reasons explained in the following article: "Epistemology Cyberattacks" https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021

ganafagol 5y ago
When appeling to authority, could you provide any sources that have not been written by yourself?
- ProfHewitt 5y ago
  The link to the article is not an appeal to authority. Instead, the article is where you can learn more about the topic under discussion. There are many references in the article that provide additional background.
- exdsq 5y ago
  In his defence, it isn't a case of the 'usual' HN appeal to authority fallacy https://en.wikipedia.org/wiki/Carl_Hewitt https://en.wikipedia.org/wiki/Carl_Hewitt
  - ProfHewitt 5y ago
    Thanks exdsq! The Wikipedia article on Carl Hewitt is way out date! And there is no way to fix it :-( Consequently, the Wikipedia article should be deleted in order not to mislead readers.
deleted 5y ago
[deleted]
drdeca 5y ago
Are you claiming that (e.g.) ZFC (which does not have orders for propositions) is not "foundations", or that it isn't consistent? Or by "foundations" are you referring to a particular system you are proposing as a foundational system, and which you have named "foundations"? You appear to justify the argument on the basis of the idea of a liar sentence. As exemplified in NFU , it is not necessary to give strict orders to things, as long as you put restrictions on how things are constructed. TST has linearly ordered types, but NFU has no need to introduce these types and orders, as just requiring that formulas be stratified is sufficient. There is no liar sentence in Peano Arithmetic. It isn't a well-formed-formula. Partitioning propositions into orders is not needed in order to prevent it being a well-formed-formula. Just, don't include anything in your rules for what counts as a wff which would let you define it. (you may object that, what if one just does the Godel numbering thing to do some quine-ing, and uses that to produce a liar sentence, but you can't express "The proposition [some number] encodes, is false" in PA (see Tarski's undefinability theorem) .) It isn't like UNK is defined in PA as "a statement UNK such that UNK iff not(provable('UNK'))". That wouldn't be a wff in PA. Rather, it is some long expression involving a bunch of quantifiers over natural numbers, and also a bunch of large numbers, and a bunch of arithmetical relations, and happens to be such that one can prove (in PA) that [UNK iff not(provable('UNK')] .
- ProfHewitt 5y ago
  Excellent questions! In order to be rigorous, a mathematical abstraction needs to be characterized up to a unique isomorphism. However, ZFC does not meet the requirement, because ZFC does not characterize sets up to a unique isomorphism. However, there is a theory that characterizes Ordinals up to a unique isomorphism that as explained in the following article: "Theory Ordinals can Replace ZFC in Computer Science" https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802 Orders can be used to block many different contradictions based on attempted recursive definitions (beyond just The Liar) as explained the following article: "Epistemology Cyberattacks" https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 The Gödel number of a proposition does not work in foundations because it does not represent the order of the proposition and because there are uncountable propositions in foundations. Criteria for foundations are proposed in the following article: "Information Security Requires Strongly-Typed Actors and Theories" https://papers.ssrn.com/abstract=3418003 https://papers.ssrn.com/abstract=3418003

ganafagol 5y ago

When appeling to authority, could you provide any sources that have not been written by yourself?

ProfHewitt 5y ago
The link to the article is not an appeal to authority. Instead, the article is where you can learn more about the topic under discussion. There are many references in the article that provide additional background.
exdsq 5y ago
In his defence, it isn't a case of the 'usual' HN appeal to authority fallacy https://en.wikipedia.org/wiki/Carl_Hewitt https://en.wikipedia.org/wiki/Carl_Hewitt
- ProfHewitt 5y ago
  Thanks exdsq! The Wikipedia article on Carl Hewitt is way out date! And there is no way to fix it :-( Consequently, the Wikipedia article should be deleted in order not to mislead readers.

deleted 5y ago

[deleted]

drdeca 5y ago

Are you claiming that (e.g.) ZFC (which does not have orders for propositions) is not "foundations", or that it isn't consistent? Or by "foundations" are you referring to a particular system you are proposing as a foundational system, and which you have named "foundations"? You appear to justify the argument on the basis of the idea of a liar sentence. As exemplified in NFU , it is not necessary to give strict orders to things, as long as you put restrictions on how things are constructed. TST has linearly ordered types, but NFU has no need to introduce these types and orders, as just requiring that formulas be stratified is sufficient. There is no liar sentence in Peano Arithmetic. It isn't a well-formed-formula. Partitioning propositions into orders is not needed in order to prevent it being a well-formed-formula. Just, don't include anything in your rules for what counts as a wff which would let you define it. (you may object that, what if one just does the Godel numbering thing to do some quine-ing, and uses that to produce a liar sentence, but you can't express "The proposition [some number] encodes, is false" in PA (see Tarski's undefinability theorem) .) It isn't like UNK is defined in PA as "a statement UNK such that UNK iff not(provable('UNK'))". That wouldn't be a wff in PA. Rather, it is some long expression involving a bunch of quantifiers over natural numbers, and also a bunch of large numbers, and a bunch of arithmetical relations, and happens to be such that one can prove (in PA) that [UNK iff not(provable('UNK')] .

ProfHewitt 5y ago
Excellent questions! In order to be rigorous, a mathematical abstraction needs to be characterized up to a unique isomorphism. However, ZFC does not meet the requirement, because ZFC does not characterize sets up to a unique isomorphism. However, there is a theory that characterizes Ordinals up to a unique isomorphism that as explained in the following article: "Theory Ordinals can Replace ZFC in Computer Science" https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802 Orders can be used to block many different contradictions based on attempted recursive definitions (beyond just The Liar) as explained the following article: "Epistemology Cyberattacks" https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 The Gödel number of a proposition does not work in foundations because it does not represent the order of the proposition and because there are uncountable propositions in foundations. Criteria for foundations are proposed in the following article: "Information Security Requires Strongly-Typed Actors and Theories" https://papers.ssrn.com/abstract=3418003 https://papers.ssrn.com/abstract=3418003