I don't know your criteria for straightforwardness, but model theory is rigorously developed, though it does use infinitistic mathematics. — TonesInDeepFreeze

What specific reduction do you have in mind? — TonesInDeepFreeze

Look at virtually any article or textbook to see that they are, modulo stylistic and symbol choices, along those lines and not with the existential quantifier in the scope of the turnstile. — TonesInDeepFreeze

Moreover, when we move on to mention 'truth', the language for PA cannot be its own meta-language. — TonesInDeepFreeze

https://math.stackexchange.com/questions/1717893/in-relatively-simple-words-what-could-be-a-model-of-sf-zfc

My question is: What does a model of ZFC look like?

The question seems to come down to a common confusion: how can a model of ZFC be a set, if we want to use ZFC to study sets?

The answer is, essentially, that things don't work that way. To study "models", we need to work in a metatheory that already has some concept of set or collection. The "metatheory" here is the theory that we use, as a tool, to study the object theory.

There are many options for such a metatheory. One option would be ZFC itself, except that (by the second incompleteness theorem), ZFC can't prove that there is a model of ZFC.

And anyway, as for pacifier, the same can be said about philosophical attempts to alleviate human suffering, from will to power, to communism, to transcendental subjectivity, to living in good faith. None of these approaches are apodictic. — ENOAH

It is logic which proves arithmetic, not arithmetic itself. — Philosophim

Moral anti-realism is pretty common these days. It is arguably the dominant, "default" position. — Count Timothy von Icarus

I am not keen on religious doctrines posing as a philosophy of consciousness, nor am I inclined to side with mysticism as anything other than a pacifier of sorts (albeit somewhat essential in its role on mental stability). — I like sushi

He proposed the project. But he insisted that all of them undertake it? Moreover, is there even one colleague to whom Hilbert insisted the colleague undertake it? — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Hilbert%27s_problems

Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne.

The other 21 problems have all received significant attention, and late into the 20th century work on these problems was still considered to be of the greatest importance.

Since 1900, mathematicians and mathematical organizations have announced problem lists but, with few exceptions, these have not had nearly as much influence nor generated as much work as Hilbert's problems.

What untested vaccines? (Of course, they're untested for the people who are taking them in tests.) — TonesInDeepFreeze

https://www.cancerresearchuk.org/about-cancer/find-a-clinical-trial/how-clinical-trials-are-planned-and-organised/how-long-it-takes-for-a-new-drug-to-go-through-clinical-trials

Drug testing and licensing

All new drugs and treatments have to be thoroughly tested before they are licensed and available for patients.

A new drug is first studied in the laboratory. If it looks promising, it is carefully studied in people. It may be then licensed if the trial shows that it works well and doesn’t cause too many side effects. You may hear this process called ‘from bench to bedside’.

There is no typical length of time it takes for a drug to be tested and approved. It might take 10 to 15 years or more to complete all 3 phases of clinical trials before the licensing stage. But this time span varies a lot.

There are many factors that affect how long it takes for a drug to be licensed.

So the existence statement is in the meta-theory, not in PA. — TonesInDeepFreeze

∃ A ( PA ⊢  ( A ⇔ ¬Bew(ÍAÎ ) ) )

PA ⊢ ∃ A ( A ⇔ ¬Bew(ÍAÎ ) )

You still don't realise that it has been proven that Gödel's version of the proof is inconsistent. — Lionino

There are people with these symptoms who are spiritual. Ergo, It's not a spiritual problem — Moliere

https://ajp.psychiatryonline.org/doi/10.1176/appi.ajp.2011.11091407

In this issue, Miller and colleagues present data from a longitudinal study of offspring from a sample of depressed and nondepressed subjects to determine if religion or spirituality influenced the onset and course of major depression over the 10 years of follow-up (1). They found, among individuals who affiliated as either Protestant or Catholic, that subjects who reported religion or spirituality as highly important were 76% less likely to experience an episode of major depression during the follow-up. In contrast, religious attendance and denomination had no impact. The protective effect was experienced primarily among subjects at high risk because their parents experienced depression.

This has nothing to do with suicide at all. — Moliere

If you've decided to end your life, for either a rational or emotional reason, that's hardly a handicap. — Vera Mont

https://www.health.state.mn.us/communities/opioids/prevention/painperception.html

The United States makes up 4.4% of the world’s population, and consumes over 80% of the world’s opioids.

That sums you up nicely. Thanks. — Vera Mont

that's right, islamic doctrine is philosophically lifeless — flannel jesus

good thing Islam isn't replying to me in this forum then — flannel jesus

I really could not care less what Islamic doctrine says — flannel jesus

Yah. I think that comes under the religious, rather the rational heading. — Vera Mont

Inescapable suffering that makes any joy in life impossible seems like a valid reason to me. — flannel jesus

Quran 4:30 And kill not yourselves. Surely, Allah is Merciful to you.

Not I, but Langendoen and Postal. If you wish you can take up the argument, I'm not wed to it, I'll not defend it here. I've only cited it to show that the case is not so closed as might be supposed from the Yanofsky piece. — Banno

https://aclanthology.org/J89-1006.pdf

This book is an extended argument in support of the theses that natural languages are transfinitely unbounded collections, that sentences are not limited in length (number of words) by any cardinal number, finite or transfinite, and that no constructive grammar can be an adequate grammar for any natural language.

https://fa.ewi.tudelft.nl/~hart/37/publications/the_papers/on_vastness.pdf

However, as I mentioned before, the authors do not so much argue for “not assuming a size law” but for “assuming the negation of a size law”. For example, the rules (if any) of English do not stipulate a maximum finite length of sentences; one can easily break such a stipulation by prefixing a maximum length sentence with “I know that”. The rules of English also do not explicitly state that sentences should be finite; one can add “All English sentence should be finite in length” to the rules or not. The authors argue, quite vociferously at times, against adding that condition mostly on the grounds that it is not a purely linguistic one. However, and this is where I disagree, they then conclude that, somehow, necessarily there should be sentences of infinite length.

https://en.wikipedia.org/wiki/First-order_logic

Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including topology and model theory.

Infinitary logic generalizes first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus, formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.

But the point is that "...the collection of all properties that can be expressed or described by language is only countably infinite because there is only a countably infinite collection of expressions" appears misguided, and at the least needs a better argument.

Your posts sometimes take maths just a little further than it can defensibly go. — Banno

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

Another important uncountably infinite set is the collection of subsets of the natural numbers. The collection of all such subsets is uncountably infinite. Now that we have these different notions of infinity in our toolbox, let us apply them to our concept of true but unprovable statements. All language is countably infinite. The set of statements in basic arithmetic, the subset of true statements, and the subset of provable statements are all countably infinite.

This brings to light an amazing limitation of the power of language. The collection of all subsets of natural numbers is uncountably infinite while the set of expressions describing subsets of natural numbers is countably infinite. This means that the vast, vast majority of subsets of natural numbers cannot be expressed by language. The above examples of subsets of natural numbers are expressed by language, but they are part of the few rather than the many. The majority of the subsets are inexpressible. They defy language.

https://math.stackexchange.com/questions/1206460/proving-that-the-set-of-all-english-words-is-countble

This is the question : Prove that the set of all the words in the English language is countable (the set's cardinality is אo) A word is defined as a finite sequence of letters in the English language.

Answer 1: There are 26 letters in the English language. Consider each letter as one of the digits on base 27. This mapping yields that the cardinality of your set is ≤|N|, hence this set is countable.

Answer 2: The set Sn of the English words with length n is finite (this is almost obvious). So it's also countable. Why is it finite? The set An of all sequences with length n made up of latin characters is finite as it contains 26n elements. Only some of these sequences are meaningful/actual English words. So Sn⊂An. So Sn is also finite. The set T for which you have to prove that it is countable is: T=S1∪S2∪S3∪... Now you have this theorem: "A countable union of countable sets is also countable". Applying it you get that T is also countable. Thus your statement has been proved.

But the whole point Wittgenstein's argument on the autonomy of mathematics systems is that a mathematical proposition is internally tied to its proof/proof system — Richard B

https://en.wikipedia.org/wiki/Model_theory

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).

If dealing with autonomous calculi then no matter how similar the rules of the two systems might be, as long as they differ - as long as we are dealing with distinct mathematical systems - It make no sense to speak of the same proposition occurring in each. The most that can be concluded is that parallel propositions occur in the two systems which can easily be mapped onto each other. — Richard B

Hence Godel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems. — Richard B

https://web.mat.bham.ac.uk/R.W.Kaye/publ/papers/finitesettheory/finitesettheory.pdf

The work described in this article starts with a piece of mathematical ‘folklore’ that is
‘well known’ but for which we know no satisfactory reference.

Folklore Result. The first-order theories Peano arithmetic and ZF set theory with the
axiom of infinity negated are equivalent, in the sense that each is interpretable in the
other and the interpretations are inverse to each other.

Perhaps the first and most obvious conclusion is that statements concerning the equiv-
alence of ‘Peano Arithmetic’ and ‘ZF with the axiom of infinity negated’ require some
care to formulate and prove. It is certainly true that PA and ‘ZF with the axiom of infin-
ity negated’ are equiconsistent for just about any sensible axiomatisation of the latter,
in the sense that interpretations exist in both directions.6 Probably this is the ‘folklore
result’ that most people remember. But for the finer result with interpretations inverse
to each other, careful axiomatisation of the set theory is required. A category theoretic
framework for interpretations is useful to direct attention to these refinements.

Why have you got it in for the math professors? — Tarskian

But why hate on the math professors? — fishfry

These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry

leaving only the beautiful sculpture that is modern mathematics — fishfry

https://en.wikipedia.org/wiki/Hilbert%27s_program

Statement of Hilbert's program

The main goal of Hilbert's program was to provide secure foundations for all mathematics.

Completeness: a proof that all true mathematical statements can be proved in the formalism.
Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
...
Kurt Gödel showed that most of the goals of Hilbert's program were impossible to achieve.

Out of the uncountably infinite and random universe of mathematical truth

Right, and despite their work being concluded for quite some time now, people several times smarter than both you and I combined still hold math and science as tools of precision and meaningful discovery. — Philosophim

I find this point more interesting. Why? — Philosophim

https://www.marxists.org/subject/marxmyths/john-holloway/article.htm

In speaking of Marxism as ‘scientific’, Engels means that it is based on an understanding of social development that is just as exact as the scientific understanding of natural development. For Engels, the claim that Marxism is scientific is a claim that it has understood the laws of motion of society. This understanding is based on two key elements: ‘These two great discoveries, the materialistic conception of history and the revelation of the secret of capitalistic production through surplus-value, we owe to Marx. With these two discoveries Socialism becomes a science. The claim that Marxism is scientific is taken to mean that subjective struggle (the struggle of socialists today) finds support in the objective movement of history. The notion of Marxism as scientific socialism has two aspects. In Engels’ account there is a double objectivity. Marxism is objective, certain, ‘scientific’ knowledge of an objective, inevitable process. Marxism is understood as scientific in the sense that it has understood correctly the laws of motion of a historical process taking place independently of men’s will. All that is left for Marxists to do is to fill in the details, to apply the scientific understanding of history. The attraction of the conception of Marxism as a scientifically objective theory of revolution for those who were dedicating their lives to struggle against capitalism is obvious. At the same time, however, both aspects of the concept of scientific socialism (objective knowledge, objective process) pose enormous problems for the development of Marxism as a theory of struggle.

Again, hyperbole. I can assure you if we were able to predict how everything in the universe worked — Philosophim

No, nothing you have discovered here has shaken the foundations of math or science. — Philosophim

What method did you use to find out that its true? — Philosophim

The case for a higher authority, an absolute authority, has to be argued philosophically. Not religiously, that is, not according anything so instantly assailable. — Constance

It sounds as though you yourself hold some rather specific and rigid beliefs that likewise are not entirely objective in their genesis. — Pantagruel

Just like I wouldn't grab a wrench if I were studying the atomic level of the universe, one shouldn't use certain language and terms when dealing with the foundations of knowledge and mathematics. — Philosophim

The hyperbole just isn't true. — Philosophim

Gödel’s famous incompleteness theorem showed us that there is a statement in basic arithmetic that is true but can never be proven with basic arithmetic. That is just the beginning of the story.

it took other people to fix the inconsistency in his proof just to then generate further issues in these updated proofs. — Lionino

In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily".

And everytime when someone makes an universal statement that ought to apply to everything, watch out! — ssu

I think you’re mis-using the word there. If everything were chaotic, nothing would exist, and if everything were perfectly ordered, nothing would change. Existence requires both. Beyond that, I can’t see the point, if there is one. — Wayfarer

Hippasus, who found irrational numbers was ostracized and when he drowned at sea, it was the "punishment of the Gods". — ssu

The existential claim carries the onus probandi — above

Out of interest, what type of believer are you? Muslim or Christian, or something less specific? — Tom Storm

Charles V's "Edict of Blood" of 1550 in the Burgundian Netherlands

No one shall print, write, copy, keep, conceal, sell, buy or give in churches, streets, or other places, any book or writing made by Martin Luther, John Oecolampadius, Huldrych Zwingli, Martin Bucer, John Calvin, or other heretics reprobated by the Holy Church.
...
That such perturbators of the general quiet are to be executed, to wit: the men with the sword and the women to be buried alive, if they do not persist in their errors; if they do persist in them, then they are to be executed with fire; all their property in both cases being confiscated to the crown.

Perhaps some can see this as chaotic, but math itself is quite logical and hence quite orderly. Unprovability or uncomputability doesn't mean chaotic. Math is orderly, we just have limitations on what to compute or prove. — ssu

https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.[2]

This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos.

Even if it obviously c is a natural number and has a precise point on the number line, not some range, we cannot prove c exactly. — ssu

The problem rises because we just assume that everything in math has to be provable. — ssu

https://en.wikipedia.org/wiki/Hilbert%27s_program

Statement of Hilbert's program

- A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.

- Completeness: a proof that all true mathematical statements can be proved in the formalism.
- Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
- Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
- Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

The existential claim carries the onus probandi (generally, existential claims are verifiable and not falsifiable, universal claims are falsifiable and not verifiable), it's not for someone else to disprove. — jorndoe

This is a far cry from the point that math can be difficult to put into words. The proof is in the very fact you're able to post online consistently for us to read your posts. That was all capable through math. — Philosophim