I am not able to establish unshakable beliefs. Are you? — Monist

I agree that mathematics rests on a set of speculative and arbitrary core beliefs, a set of axioms, which are even deemed to be circular. It is not possible to exclude logic itself from that consideration, because logic itself is also an axiomatic system. So, you cannot arrive at an unshakable belief by using logic. Science is even worse, because all its theories are deemed to have an expiry date.

In that sense, there are no unshakable beliefs unless you declare them to be such. It is your own decision what is unshakable to you. Choose wisely ...

Miracles have been claimed in every religion, society, and region. — Gregory

Ha, no, there are no miracles in Islam itself, i.e. performed by the prophet of Islam, may he rest in peace.

However, Islam refuses to comment negatively on miracles mentioned in other religions. Especially the miracles performed by Jesus are confirmed in detail and also in globo:

And We gave unto Jesus, son of Mary, clear miracles. — Quran 2:87

Im interested in some thoughts concerning how moral/immoral actions balance out. — DingoJones

If a person commits theft but regrets in it for some reason and spends the rest of their life giving most of what they have to charity (not necessarily a formal one, could just be to people he meets who are in need or whatever) then he has worked off some kind of moral debt. We might even say the person has paid their moral debt and has a surplus, moral credit, if they ended up with a huge imbalance of moral acts over immoral ones. (For example, stole a pack of gum but saved millions of lives and donated billions of dollars to charity) — DingoJones

So, can we pay off moral debt? — DingoJones

In my impression, the principle of karma in Buddhism works pretty much like you describe. It is considered to be some kind of accounting system across lives.

You suffer in this life for immoralities committed previously in this life, or in a previous life. If you commit new immoralities, you will suffer later in this life, or in a later life. Conversely, you will experience happiness later in this life, or in a later life, for your good deeds.

I am not sure that it necessarily balances out.

Your bad deeds get gradually written off as you suffer for them, but I am not sure that you can merely perform good deeds to reduce your future suffering. Perhaps, you can only write off your bad karma by suffering alone. A Buddhist monk could better clarify the details of how karma works.

Other religions simplify the same idea to merely carrying over good and bad deeds to the rest of your life, and the outstanding balance to the next life, with no reference to previous lives.

I think you would agree with me that a list of axioms does not constitute a formal system. — EricH

No, it does.

A formal system is a list of axioms.

Such formal system is always augmented with a choice of logic system, which is by default first-order logic.

A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system".[1] In 1921, David Hilbert proposed to use such system as the foundation for the knowledge in mathematics.[2] A formal system may represent a well-defined system of abstract thought. — Wikipedia on the term formal system

Since the choice of logic is pretty much always the same, it is often not even mentioned.

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems.[1] — Wikipedia on axiomatic system

In Islam, the Quran is a set of axioms while religious law ("al fiqh") is the (axiomatic) theory that emerges from the Quran. The Sunnah is considered to be already a set of theorems derived from the Quran and its use is therefore limited to a clarifying role.

Such as you talking about "statism", but being unable to define it — IvoryBlackBishop

It is a trivially simple definition. Statism is the practice of government increasingly taking over tasks that originally belonged to non-government. Simple, no?

the notion of extending the "nation, kingdom, and so on and so forth" beyond the individual, atomized "families" or "tribes" was not a recent invention at all, nor something exclusive to "secularism". — IvoryBlackBishop

Was education a government task back then? Healthcare? Dealing with marriage and divorce? I don't think so.

You also use Mafiosi who rape and "dominate" women by force and aggression as some kind of role model to champion for their supposed virility — IvoryBlackBishop

You are again trying to use some liberal-arts lying and manipulating. If I have said something, you should be able to quote it literally. Otherwise, you are just inventing that. Your strategy of lying and manipulating does not work in a written medium, because it is too easy to go back end double check. In my opinion, you are a born liar.

Oh, and I've heard of Rollo Tomoassi (which is actual the name of some obscure film character) and he's just a salesmen who isn't even doing any of the stuff he writes about, he admits himself he's "married". — IvoryBlackBishop

In what sense does that even matter? His writings seem to be influential in the "manosphere". He has a good pen. I am not going to criticize him for no reason at all. You are also a born black-mouther and bad-mouther.

All you have demonstrated up till now are truths about yourself: The fact that you routinely lie, manipulate, black-mouth, and that you are fundamentally dishonest, confrontational, and insincere.

Everything you say is contradictory — IvoryBlackBishop

No, no. You need to learn how to quote literally.

You cannot claim "you say" while not quoting what I have said. That is just a liberal-arts fraud. That may be how they taught you how to lie and manipulate, but I do not respond to that.

Because I recommended some printed books (which you may check for yourself whether they are also online or not), you infer that I don't know enough about HTML and should read a book about it? That would prevent me from ever again grievously recommending books when I don't know whether or not they are also available online? — GrandMinnow

Look, the academia are stuck in a highly inefficient way of transmitting information while charging an arm and a leg in the process. Now, this subject is not vocational at all. So, it is indeed fun as a hobby, but the only thing you can professionally do with it, is teaching in the academia, and then also get stuck in highly inefficient ways of doing things while further bankrupting an already corrupted youth. I only use online resources for my hobbies. In fact, I only use online resources for professional activities too. So, we are not going to get anywhere discussing these things any further, because we simply live in two different worlds, while I more or less despise yours. So, no, we are simply not going to do things your way, and there is no need to argue over that. We'd better agree to disagree.

What friends? You said I didn't have any. — GrandMinnow

In that case, read "HTML for dummies" alone.

You're so self-contradictory it's difficult to reply to — IvoryBlackBishop

No, that is not how to point out a contradiction.
Try again.

* ... * ... * ... — GrandMinnow

It looks like worthless dead-tree material. I don't do that. There is no longer a need to cut trees in order to publish information. There is also no need any longer to physically sit a room for that purpose. We do not need these dinosaurs. If they do not put it online, then someone will put something better online.

With a view on staying relevant, maybe you and your friends can start reading HTML for dummies?

Glad to be in that 20% then as far as that philosophy goes, wish I could say the same for you, brah, lol — IvoryBlackBishop

Well, no.

Many of these guys only stay alpha for the time that it lasts. At some point, they will just have to revert to what other men have to do, i.e. to rely on the fact that they make good money. However, they have often never invested in developing the skills to do that. So, their stories usually do not have a happy end.

It is a bit like the high-school bullies. They do well in high school, but tend to fail miserably later in life. Again, they invested in developing the wrong skills.

Furthermore, on the opposite side, most non-attached non-virgins have their very cheap moments. Just look at what they are willing to do when they run out of cash. Preferably don't film it, because that movie will not be safe for children to watch. For a little bit more cash, they will even fake their satisfaction, if that is your thing. According to the red-pill philosophy, they can no longer pair-bond. So, they mostly fake it anyway. Nothing new there.

If you cannot see through the aforementioned individuals, then yeah, then you cannot get what you want, whatever that may be.

As I have mentioned earlier, you will get much better results in every sense with someone who has a history of chastity. As I have said before, accumulating trysts with fake individuals is simply low-value behaviour.

I never payed for it — IvoryBlackBishop

Well, the general case is that you pretty much always pay for it.

Case 1. A keeper. You will end up giving her money for household expenses. In the local culture here, you even start by paying for a substantial bride gift.

Case 2. A seemingly "free" tension-relief service provider. Watch out for Weinstein-style cases. You could end up at the receiving end of a "regret" rape accusation or other back stabbing. That could go badly wrong. There really seems to be a trend to put more and more alpha players in jail. (#metoo).

In my opinion, "free" sex is somewhat an illusion. Julian Assange was supposedly also getting the sex "for free" in Sweden, and where is he now?

She's adding a constant to the LANGUAGE of the theory. — GrandMinnow

The language, the theory, and the model are separate, but related things. — GrandMinnow

In model theory, a signature σ is often called a vocabulary, or identified with the (first-order) language L to which it provides the non-logical symbols. — Wikipedia on the term signature

Yes, I see now, what Gitman wrote, is an extension of the signature, and therefore of the language of the theory. If it is a countable model then the extension will also show up in its (CS) definable language, but that is clearly not possible for nonstandard models of PA.

If you like, I can recommend introductory textbooks in this subject that will explain all this for you, step-by-step, in greater detail and context than I can give in posts. — GrandMinnow

There does not seem to be much that can be downloaded, but if you have any links to good material with examples, feel free to post them. From the paucity of materials online, the subject does not seem to be that popular, actually.

Sure, for uglier people it is, if you're a rock star or a pro-athlete (or at least look kind of like one), some women probably give it up for free. — IvoryBlackBishop

Well, according to the red-pill philosophy only 20% of the men are alphas, i.e. men with whom available women are happy to have sex for free. Most men, 80% of us, are not alphas. I do not consider myself to be an alpha either (in terms of handsome looks or celebrity status). Still, that does not matter, because a beta can trivially achieve the same results as an alpha by paying out relatively small nominal fees. Just make sure to really "exchange". Never give something for nothing. "So, ok, I will fix your car or your computer, or whatever, but what am I getting in exchange?" If she is giving it away for free to alphas, why even give her a slice of your pizza for free? "So, you urgently need $20? Then work for it. You can duly sweat on my next tension-relief gig!" "Beta orbiters" are guys who simply don't get that.

I fail to see your point, as usual. — IvoryBlackBishop

Well, you seem to have no experience with different kinds of society. That is undoubtedly why.

Get to know people who live in other areas of the world and you may understand how these things work in their environment. I have rapidly adjusted to SE Asia, hitting the ground running. I do understand the inter-gender dynamics here, if only, because I definitely manage to get the quality results that I want. The proof is always in the pudding. ;-)

A model is not a language. — GrandMinnow

Well, Gitman clearly mentions that she considers a model to be (some kind of) language:

slide 10: "Expand LA by adding a constant c to obtain the language LA∗ = (+,·,<,0,1,c)."

So, that is why I first said something similar: "Just add a Löwenheim-Skolem ω symbol to the model's language."

It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. — Wikipedia on Löwenheim-Skolem

These infinite cardinals are clearly an element in Cantor's $\aleph$ (or $\beth$) sequence.

Therefore, Gitman's "language" does not always have a countable number of sentences. So, what kind of "language" is it supposed to be?

I've never "payed" for it — IvoryBlackBishop

"Free" is never really "free" when it is about sex. The other side usually has possibly hidden expectations. When these expectations are not fulfilled, the other side may very well try to get back at you. Just look at the Weinstein case. All these women slept with him hoping that he would land them an acting gig as an actress. I am sure some of them did get what they wanted, but the ones who didn't now scream "rape!". This problem rarely occurs if you finish such "casual sex" episode by paying a nominal fee.

I've had a few lonely housewives preposition me — IvoryBlackBishop

That can go even more badly wrong, because in that case, there is even a second party who could possibly take an interest in such episode. If he figures you out, he may detect a vulnerability to exploit, and in that way get even with you. In my opinion, it is just not worth it. With the large number of available service providers -- every non-attached non-virgin is basically for rent here -- I don't even see why the juice would even be worth the squeeze? Furthermore, islamic-law advisories utterly condemn that kind of behaviour, i.e. sleeping with someone else's wife.

You contradict yourself, you're now saying men should be "civilized, effminiate, and monogamous" — IvoryBlackBishop

Not at all.

First of all, it is about understanding that a man should not even want most women. I certainly don't. They are simply not fit for purpose in a more or less standard breeding strategy. They could possibly still be fit as "tension relief" service providers, paid or free, but if you organize your personal life properly, why would you even need those?

Furthermore, I have never excluded polygamy as as breeding strategy.

men who chase after any woman they want. — IvoryBlackBishop

In my opinion, if you are "chasing", you are probably doing it wrong.

Say that there are two cases.

Case one. You just want to get some "tension relief". In that case, it is a case of supply and demand. No need to chase. If you pay slightly more than the market price, the service providers, including the so-called "free" ones, will all be queuing to work on the gig for you.

You may only need to chase, if you want the "tension relief" service for free. Since I make way more money by working one hour than the amount that I would save by chasing free service for one hour, I think that such strategy is absurd. Middle class girls (and up) may only want to do "free" casual sex, but everybody else on lower income cannot really afford that. They will prefer monetary compensation. Note that virgins will never do "tension relief" gigs, because they can still get a much bigger bride gift ("mahar"). Again, I believe that it is better to organize your personal life in such a way that you do not even have any need for the occasional "tension relief".

Case two. You have detected a possible "keeper". Again, no need to chase. Tell her you are possibly interested in paying her bride gift ("mahar"). That is usually substantially more than just a nominal amount of money, and there are rarely many candidate suitors who are capable and willing to pay it. Therefore, you will often be the only option in town for the foreseeable future. Therefore, unless she has another interesting deal already in the making that is about to be closed, you can expect red-carpet treatment from there on. Whenever I talk about possibly "keeping" her, even just jokingly, the usual reaction is: "So, when are we beginning? It is taking way too long for you to finalize!"

I've been with somewhere between 15-20 women, nor was I ever married. — IvoryBlackBishop

Cool. You're point is? — IvoryBlackBishop

In terms of biology, I consider that to be low-value and even possibly worthless behaviour, and certainly not something to brag about. I do understand why a man could discretely use female "tension-relief" services against a fee, or even for free (=more dangerous), because of his current circumstances, but why boast about that? It is very akin to going to the toilet with a view on producing some bowel movements. Does it make sense to brag about how many times you may have done that today?

She doesn't preclude uncountable languages in general. But where she proves the existence of non-standard models of PA, the languages mentioned are countable languages. Uncountability does not play a role in it. — GrandMinnow

This is the point where I somehow backtracked on what Gitman wrote, while I initially thought it was actually really good. I really ran off with the idea that a model is a language. It clarified the difference between a theory, i.e. a set of axioms, and a model, i.e. a CS-like grammar describing sentences. I still think of things like that for countable models, but I have to leave it open if the concept properly applies to models that are larger than that, such the nonstandard models of PA.

Countability is often assumed to be an inherent property of any language in other arguments such as in Richard's paradox:

Thus there is an infinite list of English phrases (such that each phrase is of finite length, but the list itself is of infinite length) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length lexicographically (in dictionary order, e.g. we can use the ASCII code, the phrases can only contain codes 32 to 126), so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1. — Wikipedia on Richard's paradox

The argument is clearly very similar to Cantor's diagonal argument. In the explanation above, language itself is clearly considered countable.

I think that it would be useful for Gitman to explain why such uncountable nonstandard model would still be a legitimate language.

I currently see it more as some kind of dense cloud, centred around a choice for ω, of number symbols/strings, that cannot be adequately captured by the concept of language.

I've never been in anything which would amount to "violent conflict" — IvoryBlackBishop

For now.

I've been with somewhere between 15-20 women, nor was I ever married. — IvoryBlackBishop

In our species, mankind, that kind of behaviour is widely considered to be mostly a waste of time. It is understandable behaviour but it is nevertheless quite useless, because it is not particularly productive.

Our species' mating and breeding strategy is very similar to a good number of bird species, such as cockatoos, where the female stays in the nest, sitting on the eggs, and later, guarding the chicks, while the male flies out to find fruits and nuts to be regurgitated in the nest.

The cockatoos' incubation and brooding responsibilities may either be undertaken by the female alone in the case of the black cockatoos or shared amongst the sexes as happens in the other species. In the case of the black cockatoos, the female is provisioned by the male several times a day. — Wikipedia on Cockatoo breeding

Pretty much every religion insists on the idea that useful sex is part of the overall breeding strategy.

I look after my three children here. I bring "the fruits and the nuts", pretty much in accordance with the basic biology of humanity and in line with Islamic-law advisories.

I consider the proper breeding strategy not to be about merely sleeping with arbitrary females. It is not that "pumping and dumping" would be hard to do here in SE Asia. Especially in Vietnam, there is a specialized class of young women doing that in exchange for not much money. So, it is certainly possible to use that kind of services for convenient "tension relief" but on the whole this behaviour can be deemed biologically low-value or even worthless.

Spent 2 months in Siem Reap a few years ago. Probably my favourite place. — Michael

Agreed. It is great!

Phnom Penh and Kampot are also superb. Fifteen minutes across the Vietnamese border from Kampot, you are in Ha Tien, Vietnam, which is a nifty provincial town with fantastic seafood restaurants, not to mention, dirt cheap. There's a ferry from there to Phu Quoc island, which lots of people say good things about, but I haven't crossed over yet. It is somewhere on the agenda, though!

Whereabouts? Went to Thailand recently. Loved it and want to move back there. — Michael

1. Cambodia
2. Vietnam
3. Philippines

All three countries are fantastic in their own special way. I rarely hang out in Thailand, though. The last time was in 2018 for two weeks in Chiang Mai. Thailand is probably an ok country but it does not "madly" attract me. Maybe it lacks that little small bit of authenticity to make it addictive! ;-)

Fact is those groups want "legal" rights. — IvoryBlackBishop

Not true. MRA do, but MGTOW don't.

MGTOW are simply bailing out. They do not ask for anything to change. It just reflects the growing trend of men bailing out from the workforce and out of "relationshits". They just don't want to be someone else's plough horse on the "plantation".

It is not possible to change anything to society for MGTOW to change their mind. They have decided to move on, and they are simply not coming back.

Given the fact that I also believe that western society is beyond salvation, I agree with MGTOW and not with MRA, whose ambitions I consider to be pointless. I love it here in SE Asia. I am also not coming back. Ever.

Says the guy talking about MRAs and whatnot, when historically, as far as ancient cultures go, MRAs and MGTOW wouldn't have any "rights" to begin with or petition for; do you think an ancient monarch would tolerate him — IvoryBlackBishop

MRAs are not the same as MGTOW.

Furthermore, rights are not something the monarch gives you but something that you extract at gunpoint. Hence, it is mostly a question of who defeats whom in battle.

Sorry - I wasn't clear. What I'm looking for is some axioms of Islam. — EricH

Every verse in the Quran.

(https://quran.com)

You really should stop generalizing. Secularists-atheists by definition do not necessarily try to replace anyhing. — Nobeernolife

Activists hold placards during a protest demanding civil marriage in Lebanon. There is currently no Lebanese civil personal status law.

Lebanon does not have a civil code regulating personal status matters. Instead, there are 15 separate personal status laws for the country’s different recognized religious communities including twelve Christian, four Muslim, the Druze, and Jewish confessions, which are administered by separate religious courts.

Religious authorities often promoted this judicial pluralism as being essential to protecting Lebanon’s religious diversity. In reality, the multiplicity of laws means that Lebanese citizens are treated differently when it comes to key aspects of their lives, including marriage, divorce, and custody of children.

This variation has prompted rights activists in Lebanon to advocate for civil personal status law that would guarantee that citizens are treated equally, while ensuring that their freedom of belief is respected. — Unequal and Unprotected. Women’s Rights under Lebanese Personal Status Laws

They do.

Atheists are hellbent on imposing their views onto others under the nomer of "human rights".

So, these "activists" want "to guarantee that citizens are treated equally".

Their idea is to replace religious law by their own secular inventions. Of course, that will just lead to a western-style divorce-rape system where men will no longer want to marry, and to a collapse in the fertility rate. It means the end of the nuclear family. The religious communities do not want their law to be replaced by something that is known not to work.

Marriage is meaningless anyway for atheists. So, why do they even bother?

The guarantors of the current system of religious marriage and divorce are the guns, cannons, and mortars of Hezbollah. If these "activists" try to push ahead with their plans, it will lead to violent combat. So, in a sense, I think that it is still a good idea that they try, because it will thoroughly weed out and decimate their demographic. There are just too many arrogant atheists in Lebanon, and violent conflict will duly solve that problem.

First, atheism is simply a personal position on one issue, and not a belief system, so there is nothing to rely on. — Nobeernolife

It is about what the secularists-atheists try to replace religious systems by. It is exactly because their views are not a legitimate system and just some kind of vague personal position that I am so opposed to their social "solutions".

Secondly, there are plenty of religions that do not come with a "religious law" — Nobeernolife

Well, that is not completely true either.

When push comes to shove, you will see that e.g. a Christian community suddenly also has a religious law. They certainly had one in the Ottoman empire, and they still have one in Lebanon today, where the fifteen religious communities happily administer their own marriage and divorce regulations.

Islam: Sunni, Shia, Alawi

Christianity: Maronite Catholicism, Greek Orthodox Church, Greek Catholicism, Armenian Orthodox Church, Armenian Catholicism, Syriac Orthodox Church, Syriac Catholicism, Protestantism, Chaldean Orthodox Church and Catholicism, Other Christian Denominations.

Druzism, Judaism — Fifteen communities administering their religious law in Lebanon

The same can be said of Hinduism. They did not have a religious law until they argued that they actually had one:

Dharmaśāstra became influential in modern colonial India history, when they were formulated by early British colonial administrators to be the law of the land for all non-Muslims (Hindus, Jains, Buddhists, Sikhs) in South Asia, after Sharia i.e. Mughal Empire's Fatawa-e-Alamgiri[12][13] set by Emperor Muhammad Aurangzeb, was already accepted as the law for Muslims in colonial India.[14][15][16] — Wikipedia on Dharmaśhāstra

So, the antinomian idea that a religion has no religious law gets abandoned rather swiftly when they are asked to self-govern the personal and family law of their religious community. Then you will find that they suddenly re-discover their own religious law.

In fact, I find all fifteen codexes in Lebanon quite reasonable, much more so than the completely failed approach in secular western countries with imploding marriage -and fertility rates. The proof is always in the pudding.

Secularist. You rail against secularism. — fdrake

And against statism too.

Now, I do not believe at all that atheists should become religious or adopt the self-discipline of religious law. They can obviously do what they want.

It is just that the western-style state is way too intrusive in people's lives. They even try to substitute community charity (zakaat and sadaqah) by financially bankrupt social-security statism.

It really isn't. "Left liberal", "social democrat", "progressive liberal" are much better. — fdrake

According to the conspiracy theory, which emerged in the late 1990s, the Frankfurt School and other Marxist theorists were part of a conspiracy to attack Western society by undermining traditionalist conservatism and Christianity using the 1960s counterculture, multiculturalism, progressive politics and political correctness.[50][51][52] — Wikipedia on cultural Marxism

I use the term "cultural Marxism" as a blanket reference to the vague collection of ideologies that emerges from atheism and that rejects the requirements of traditional religious law. I don't know of a better term to capture the idea.

I do not count multiculturalism under that nomer, because I do not see that as a problem.

For example, the Ottoman empire was staunchly multicultural and multi-religious. The Ottoman millet system worked fine for close to a millenium. There is absolutely no need for everyone to share the same culture or the same religion.

When people reject religious law, they will still organize society according to a particular model. Whether they see themselves as leaning "left" or "right", the result will fundamentally be the same.

It will consist of a Statist land grab in which governments repossess solidarity from extended family, clan, tribe, and religious community through state-run social security, while also bringing education and healthcare under secular, atheist state control. By peeling off layer after layer of the onion of traditional social structure, the Statist land grab will eventually also reach the nuclear family, and rip it apart, making generational reproduction impossible in the process. The individualized and atomized person in such society will ultimately even fail to survive. At that point, cultural Marxism will have destroyed itself.

Both political "left" and "right" are rife with Statism and hellbent on new Statist land grabs. That is the "Marxist" part about it. In the end, it is not about the economy, because their attack on human society is directly aimed at the social structure, i.e. its wider notion of "culture".

Can you give me a few axioms - in plain language if possible? :smile:

If there is a more comprehensive list available - again with plain language explanations if available - you can give me a link to check out. — EricH

In mathematics it is relatively easy to find the basic rules of a theory structured as axioms. The following are the dominant axiomatizations in mathematics:

• Axioms of standard number theory (PA)
• Axioms of standard set theory (ZFC)

There are many more. The following is the dominant axiomatization of logic:

• Axioms of propositional logic

The axiomatization of logic gives the basic rules for the standard formal language used by the other mathematical axiomatizations (There are other formal languages possible). Hence the quite special status of logic in mathematics and in other axiomatic disciplines. The standard formal language in mathematics is first-order logic.

You can find lots of examples of various axioms in these three standard axiomatizations. Simple example of an axiom:

5 + 3 = 3 + 5

You can swap the order of operands in a sum. Generally:

a + b = b + a, for any a,b being a natural number.

Standard number theory will define what the term "natural number" means, up to final interpretation by an actual natural-number model. A model is also called: an interpretation, a universe, or a world.

The standard model of natural numbers is a regular language such as for example:

$\Bbb{N}$ = 0 | [1-9] [0-9] *

along with the symbols (+, x).

You can pick any number of arbitrary, distinct symbols $(α_0, ..., α_n)$, two or more, in order to define a collection of strings that represent the natural numbers:

$\Bbb{N}$ = $α_0$ | [ $α_1$-$α_n$ ] [ $α_1$-$α_n$ ] *

along with two arbitrary, distinct symbols (⨁,⊗), that will satisfy the standard theory of natural numbers (PA).

If you get used to the terminology, it can be relatively easy to learn to think in terms of axioms. Outside mathematics, axiomatic systems are also possible but have never been seriously formalized. So, their nature may be clearly axiomatic but they do not use symbolic language to express themselves.

I don't think their mindset is that deep about it, no. — IvoryBlackBishop

Rollo Tomassi is quite deep. His website, the rational male, certainly is. There are actually quite a few of these websites. They also have elaborate youtube and bitchute channels. If you want to figure out what they think, you will quickly see that they specifically take issue with exactly the two problems that I mentioned.

I think rather than mindlessly "railing" against the system as a whole, using sensationalized words like "gynocentric", one should give specific examples of laws which should be changed on a state or federal level, and something akin to an actionable plan for or on how to do it. — IvoryBlackBishop

That is pretty much also the position of the MRA (Men's Rights Activists).

The MGTOW crowd rejects that point of view. They do not want to change the system.

MGTOW prefer to completely withdraw, and just let the system collapse, because they consider the system beyond salvation already. I think that they are also right on that matter. There is no point in fixing anything. Just let the laws of nature take over instead. Furthermore, it is enough that men withdraw in droves from the labour force and from relationships to effectively precipitate the inevitable collapse. I did the same. I cannot be bothered about failing systems because I live elsewhere now.

Are you going to move to a strict Sharia law nation? — IvoryBlackBishop

Islamic law is first and foremost a matter of self-discipline. There is some need for a government to do conflict resolution and victim compensation but that is for me personally not a pressing matter on a daily basis. Hence, an Islamic government is merely nice to have. A hands-off government is actually equally good.

I don't think Islam would be too keen on "MGTOW" men deciding to devote their life to porn and anime masturbation either — IvoryBlackBishop

We don't know if a particular person is devoted to porn and anime masturbation. If it is an offence against Islamic law, which is possibly the case, it would still require the quorum of witnesses before it would be actionable.

Furthermore, what conflict would there need to be resolved or what victims would there need to be compensated in the realm of porn and anime masturbation? Hence, all you could ever see about that is a religious advisory decidedly recommending against porn and anime masturbation, prominently affixed to a virtual wall on the internet. Case closed.

I suppose that if two people enter a relationship, there is always a "possibility" like that in mind, but regardless, I do not think that an attitude of cynicism, nihilism, or paranoia is healthy, and If anything just fodder "trash TV" shows like Jerry Springer or Steve Wilkos. — IvoryBlackBishop

Demanding chastity before agreeing to marriage is not "an attitude of cynicism, nihilism, or paranoia". For Muslims, it is a non-optional requirement. Since MGTOW come to exactly the same conclusion as Islam, I cannot possibly criticize MGTOW, and certainly not on their advisories. In fact, the ulema (the Islamic religious scholars) have been issuing exactly the same advisories for decades now.

When I look at MGTOW, each one of their concerns and each one of their solutions comply with Islamic law. Hence, no bad word from me on MGTOW.

In contrast, the worldview of the "MGTOW/Incel" is the polar opposite, one of pop nihilism, hedonism, materialistic consumerism, and so on — IvoryBlackBishop

I resolutely refuse to criticize MGTOW men. They do not want to have relationships because of the gynocentric court system and because they believe that society has thoroughly perverted the opposite sex.

First of all, in Islamic terms, if marriage and divorce are not governed by Islamic law, then the man is advised not to enter the arrangement. Secondly, chastity is indeed a very non-optional requirement:

You may marry the chaste women among the believers, as well as the chaste women among the followers of previous scripture, provided you pay them their due dowries. — Quran 5:5

The Quran simply does not want a man to enter a marriage arrangement when there are serious questions or doubts about chastity. Therefore, the religious advisory en provenance from the Quran is that the MGTOW guys are absolutely right on both counts.

Personally, I fundamentally solved the problem by emigrating to a different jurisdiction, in a galaxy far, far away (SE Asia) and by being absolutely paranoia, from a non-negotiable position, concerning the chastity of my spouse. If I had not emigrated, I would also have been staunchly MGTOW today.

I didn’t say there is anything wrong with Gitman. You seemed to have skipped what I said about it. Actually, I should be sharper by saying that it appears to be a step in proving existence - and, again, an actual non-standard model of PA is a model for the language of PA, not a model for a different language with an additional symbol. — GrandMinnow

She extends the language of the model in slide 10: "Expand LA by adding a constant c to obtain the language LA∗ = (+,·,<,0,1,c)."

The term "language" in her slides is something that can have an uncountable number of sentences. Her model cannot have a CS grammar, unlike the standard model, because CS grammars can only represent a countable number of sentences.

And by w-1, do you mean some kind of ordinal subtraction, or something else? — GrandMinnow

Ask Victoria Gitman about what she means with that term in her slide. I think that I understand it, but better ask her instead.

It’s not vitriolic to point out that you are posting misinformation ... — GrandMinnow

Since you are not much useful in correcting errors, besides merely criticizing other people, your contribution is simply worthless and just annoying. I am not an expert on model theory. I just gave my understanding on something it investigates. What you are doing, however, is completely worthless. In what sense do you believe that your vitriolic remarks would be helpful in any way? If they successfully prevent people from discussing the topic, will you have achieved your nefarious goals? You are not here to discuss anything. You are here to point out that you believe that you know the subject better than anybody else here, mostly by putting them down. Don't you see that nobody needs or even appreciates your presence? Your obnoxious attitude turns you into a worthless individual whom nobody likes to be around.

Am I supposed to be offended by this extreme-right, neo-Nazi conspiracy-theorist label? — Possibility

I like the term "culturally Marxist" because that is exactly what it is. I don't care who else adopted the term. If the Neo-Nazis adopted the term "yellow beef broth" would I stop using it? ... not really.

My experience with both single-sex and co-ed schooling prompts me to seek the co-ed option for my children, without a doubt. ... Fear of our own ‘sexual urge’ is like the tail wagging the dog. ... Yes, puberty complicates the education environment and increases risk ... — Possibility

If you think that co-education is good for your children, then go for it. What else can I say?

No, you have it very wrong. The universe has additional members. We don't add symbols that engender sentences not already in the theory. — GrandMinnow

I asked you what is wrong with Victoria Gitman's slides.

And I have no idea what you mean by w-1 and w-2 in the context of saying w-1, w-2, w, w+1, w+2. — GrandMinnow

You can find that literally in her slides. Did you even look at them?

A non-standard model of PA I guess can be visualized through the method Gitman mentions, but literally a model of PA is itself a model for the language of PA, not for a language with an added symbol. The elements with 'c' in her slides are just elements of the universe. — GrandMinnow

I like her slides.

Unlike your vitriolic remarks, they add to the understanding of the topic. In fact, I do not care about what you may think about her slides, because she turns out to be useful while you are not. Her representation of ... ω-2, ω-1, ω, ω+1, ω+2, ... is also quite clarifying, unlike what you are doing.

You are completely confused about this and other topics in mathematical logic. Please stop and instead first read an introductory textbook in the subject. — GrandMinnow

It is my OP. I posted the topic. If you do not like it, feel free to unceremoniously fuck off, will you?

Fitrah is an axiom. — EricH

No, it is a philosophical explanation for why we adopt axioms. It is not an axiom itself.

Adding a symbol is not relevant, not due to whatever you said about sizes of models, but rather more simply that it is not even involved in the notion of a non-standard model. — GrandMinnow

A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934). — Wikipedia on construction of nonstandard models

This is typically represented by adding a symbol ω. If such model were of countable size then adding that symbol would be enough to represent these additional elements as: ... ω-2, ω-1, ω, ω+1, ω+2, ...

In her slides, Victoria Gitman explains this literally as:

slide 10: "Expand LA by adding a constant c to obtain the language LA∗ = (+,·,<,0,1,c)."
slide 11: "M has c+ 1,c+ 2,c+ 3 ,... as well as c−1,c−2,c−3,...."

So, what is wrong with her slides?

Again, as I have said already, it would be enough to expand the grammar of the model with a constant, if the size of the model remained countable. It does not work for larger models because grammar cannot represent uncountable models.

That's computer science, or a special bland of computer science and model theory that you are working out live? — GrandMinnow

Look, if you do not want to make constructive remarks, then why do you make any remarks at all?

Then it's model theory. And your statement about it is plainly incorrect. — GrandMinnow

Many infinite-size models cannot be handled by any grammar, because their sentences are necessarily countable, while these models may not have a countable cardinality. It can only work for something like ZF-inf, of countable cardinality.

So, no, I actually agree, throwing that symbol into the fray, won't make it work.

Much better to figure it out by reading a textbook that develops the concepts and terminology systematically, rather than posting misleading confusions for other people to read. — GrandMinnow

The confusion is merely in the lack of formality; but that was actually expected by the reader. Furthermore, what you write, does not necessarily help anybody to understand the subject either.

This is a notion of non-standard model not in model theory (in mathematical logic) but in computer science? — GrandMinnow

That would be impossible in computer science.

Computer science can fundamentally not handle infinite cardinalities, because they are generally not computable. They are sometimes handled symbolically, though.

In computing, NaN, standing for not a number, is a member of a numeric data type that can be interpreted as a value that is undefined or unrepresentable, especially in floating-point arithmetic. Systematic use of NaNs was introduced by the IEEE 754 floating-point standard in 1985, along with the representation of other non-finite quantities such as infinities. — Wikipedia on NaN

I do not believe that there is any guarantee that this NaN symbol makes sense in a model-theoretical sense. These symbols may show up, but it really depends on the implementation standard what exactly they mean. Furthermore, that symbol is mostly considered to be the result of a bug, which should not even show up ...

If your usage is is indeed standard computer science, then, yes, the terminology is radically different from basic mathematical logic. If you insist on using the terminology differently from the way it is used in ordinary discussions about hereditarily finite sets and PA, then you need to clearly state your system of terminology from your own basics rather than, without due specification, mixing it up with ordinary usage. — GrandMinnow

I only get to figure that out by actually doing it. If I don't do it, then I will never figure out what the problems are, not even the problems with the terminology.

Furthermore, I do not investigate all topics in computer science, many of which are not necessarily interesting to me. An example of how the term "formal language" may appear in this realm:

Automata theory is closely related to formal language theory. An automaton is a finite representation of a formal language that may be an infinite set. Automata are often classified by the class of formal languages they can recognize, typically illustrated by the Chomsky hierarchy, which describes the relations between various languages and kinds of formalized logics.

Automata play a major role in theory of computation, compiler construction, artificial intelligence, parsing and formal verification. — Wikipedia on automata theory

The meaning of "formal language" in automata theory is not necessarily the same as in mathematical logic:

For instance, a language can be given as:
those strings generated by some formal grammar;
those strings described or matched by a particular regular expression;
those strings accepted by some automaton, such as a Turing machine or finite-state automaton;
those strings for which some decision procedure (an algorithm that asks a sequence of related YES/NO questions) produces the answer YES. — Wikipedia on formal languages

In fact, now I see that I should probably use the term "word" instead of "sentence" for natural numbers, but also not necessarily, because in terminology compiler construction they would still be sentences ("The automaton accepts a sentence"). The term sentence in mathematical logic is probably the most closely related to "those strings for which some decision procedure produces the answer YES".