the difference in meaning that AC has in the two different systems — sime

Of course, semantics for intuitionistic systems are different from semantics for classical systems. But the question of equivalence is that of derivability.

The axiom of choice holds trivially as a tautology in sets constructed in higher-order constructive logic — sime

Reference please.

So one could even say that absence of LEM implies AC (or perhaps rather, that AC is an admissible tautology in absence of LEM). — sime

That makes no sense. 'Implies' means proves in this context. And one cannot prove what is otherwise not provable by weakening the proof logic. (But I can't opine on 'admissible tautology' since I don't know your definition.)

when speaking of AC not in the sense of an isolated axiom, but in the commonly used informal vernacular when speaking of choice principles in their structural and implicational senses — sime

AC is an exact formulation. It is not expressed as "commonly used informal vernacular when speaking of choice principles in their structural and implicational senses", whatever you might exactly mean by that.

There are different formulations that may have equivalences, and there are complications throughout, but I know of no proof nor mention in the article you cited that shows the equivalence of AC with LEM in intuitionistic set theory. The SEP article does say "each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is equivalent (in intuitionistic set theory) to a suitably weakened [italics in Bell's earlier article] version of the axiom of choice. Accordingly these logical principles may be viewed as choice principles." But the question was not that of various choice principles but of AC itself, and we have not been shown a proof that AC and LEM are equivalent in intuitionistic set theory.

Explicitly constructive mathematics goes back at least a hundred years, and with roots in the 19th century too. It has great importance toward understanding foundations. I think interest in it goes well beyond any need for assigning research topics.

that is true for ZF, since it is built upon classical logic — sime

That is not the reason. The reason is that LEM does not imply AC, whether with intuitionistic or classical logic.

intuitionistic set theory, in which choice principles and LEM are approximately equivalent as documented in the SEP article on the axiom of choice. — sime

I looked at that article briefly. I did not see mention of an "approximate" equivalence.

Whether classically or intuitionistically, there is not an equivalence. Rather, there is only the one direction: AC implies LEM. But the other direction that is needed for equivalence - LEM implies AC - is not the case.

the axiom of choice is equivalent to the law of excluded middle — sime

That is not correct. It is the case that Z (even without the law of excluded middle (LEM)) and the axiom of choice (AC) together imply LEM. But it is not the case that Z (which includes LEM) implies AC.

captures the essence of Cantor's views on infinity. — TheMadFool

Sure, if "captures the essence" means grossly mischaracterizes with ignorant confusions.

If you think there is anything wrong in my rebuttal, then you should be able to point to it exactly.

I asked you what version of Godel's proof have you read in a paper or book. That is, what writing did you base your previous post on?

The choice of number 2 is irrelevant to my argument. — TheMadFool

I pretty much figured that you didn't know what you were writing when you said that there is a 1-1 correspondence.

The analogy is perfect.
— TheMadFool

You still have not addressed my rebuttal.
— TonesInDeepFreeze

The analogy is perfect [...] — TheMadFool

I pointed out that you have failed to address my rebuttal. And your reply to that is to again fail to address my rebuttal.

I'm fairly confident that what I wrote would've brought a smile to his face. — TheMadFool

Yes, it's possible he might get a chuckle at your hapless ignorance.

He was a deeply troubled man I believe, in no small measure due to Leopold Kroenecker's scathing criticisms of his life's work. — TheMadFool

It wasn't just that Kronecker criticized the work. But it does seems reasonable to think that his professional difficulties vis-a-vis Kronecker might have contributed to his poor mental condition, but I don't think we know for sure.

That is is nothing like Godel's proof. On so many levels it is nonsensical.

What actual version of a Godel's proof have you read in a paper or book?

There's a precise 1-to-1 correspondence between 2 and K — TheMadFool

Incorrect.

2 = {0 1} and has cardinality 2.

K = {K} and has cardinality 1.

You haven't addressed it — TheMadFool

If you link to where you first posted it, then I'll link to where I answered it.

The analogy is perfect. — TheMadFool

You still have not addressed my rebuttal.

he defines infinity as a set whose members can be put in a 1-to-1 correspondence with the set of natural numbers. — TheMadFool

I don't know how he reads in the original German, but the above is not how the set theory that came from Cantor works.

We don't define "infinity" as a noun. Rather, we define the predicate 'is infinite'. And the definition is NOT

x is infinite iff x is 1-1 with N.

Indeed not, since there are infinite sets that are not 1-1 with N.

Rather, the definition is:

x is finite iff x is 1-1 with some natural number.

x is infinite iff x is not finite.

An alternate definition is equivalent to the above with the axiom of choice:

x is infinite iff there is a proper subset s of x such that x is 1-1 with s.

Please address the proof. I'll restate it here. — TheMadFool

Since it's a restatement, I don't need to address it again, since I've replied to your "proofs" already, in quite detail. And you have not gotten back to me on my replies. Typically, all you do in reply is to restate your incorrect argument that had just been refuted.

Moreover, I just gave you a proof in my last post, and you have not addressed it.

Suppose K = {K}. Let ~x=K and ~xeK. Then ~ {x K} = K but K e {x K}. — TonesInDeepFreeze

[the analogy] makes so much sense. — TheMadFool

I gave you specific detail why the analogy doesn't work for you. Instead of responding to that, you merely reiterate your claim that you are right.

That entire passage is merely a report of notions and terminology of mathematical logic. It's nowhere even close to a philosophical statement. Except the last sentence, which does have a philosophical aspect. And philosophically it is very little - certainly not a philosophical stance and certainly not "dogmatic".

In that last sentence, I merely say that it seems to me that when we make correct computations in arithmetic, we can take the results as true. To say this more fully: Such computations may be reduced to primitive manipulation of such things as, say, plain tally marks - the most simple, most direct mathematical reasoning that I personally can imagine. Put another way, this is merely clerical attention to mechanical procedures. Now, if someone wants to express extreme doubts of computational arithmetic, then I would say, "If you think we are not justified in accepting truth from even the most simple results of manipulation of tally marks, then what mathematical knowledge do you think is justified?" I don't even claim that the person would not have a satisfactory answer. I only say that I personally don't know of one.

That is quite on the exact opposite end of the spectrum from dogmatism.

You understand what "inherent" means don't you? — Metaphysician Undercover

Your boorish condescension is stupid.

The question is whether or not it is possible for a set to be free from inherent order, i.e. having no inherent order, as fishfry claimed. You still don't seem to be grasping the issue. — Metaphysician Undercover

I never said that the set of orderings of a set is not inherent to the set. I said over and over and over that sets have multiple orderings. The point I have been making to you is that you have not defined what it means for one of those orderings in particular to be "THE inherent ordering". You are the one who doesn't grasp the issue.

There are multiple orderings. Given a reasonable sense of 'inherent', the orderings of the set are all inherent to the set. In set theory and abstract mathematics. EVERY property of an object is inherent to the object. (Mathematical) objects don't change properties. They have the exact properties they have - always - and no other properties - always. There is no "time" operator that allows (mathematical) objects to have different properties at different times, so the properties are inherent. That is just a report on set theory, which doesn't have a "time" operator.

But the point you keep missing is that you have not defined what it means to say that one of the orderings in particular is "THE inherent ordering". They are all orderings of the set, and they are all inherent to the set. I have put 'THE' in all caps about a hundred times now. The reason I do that is obvious, but you still don't get it.

This started with discussion of the axiom of extensionality. With that axiom, sets are equal if they have the same members. In that regard, a set is determined by its members, whatever the set of orderings of the set might be. And, of course, for every set there is the set of all the orderings on that set. That set of all the orderings on the set is "inherent" in the sense that it doesn't change. But the point you don't get is that there is not one of those orderings that is in particular "THE inherent ordering" while the others are not. And it seems the reason you don't get that is because you started out needing to deny the sense of the axiom of extensionality itself, even though you are ignorant of what it does in set theory and you are ignorant of virtually the entire context of logic, set theory and mathematics.

I cannot say what the inherent order is, for the reasons explained. Do you have a problem with those reasons? Or do you just not understand what I've already repeated? — Metaphysician Undercover

I have rebutted great amounts of your confusions. You either skip the most crucial parts of those rebuttals or get them all mixed up in your mind.

Anyway, to say that there is "THE inherent ordering" of a set, but not be able to identify it for a set as simple as two members is, at the least, problematic. But more importantly, you cannot even define the "THE inherent ordering" as a general notion. That is, you cannot provide a definition like:

R is the inherent ordering of S if and only if P

where P is the definiens.

And your notion is so ridiculous that you say that if one did attempt to identify "The inherent ordering" of a set then one would not correctly choose "The inherent ordering". What? For a set with two members, I have a 50% chance of identifying "THE inherent ordering" (if there were such a thing) just by guessing.

I changed my mind on that days ago — Metaphysician Undercover

Do you mean this post?:



Whatever "change of mind you had" in that pile of confusions, you said inter alia:

(1)

Order is a spatial-temporal concept [bold added] — Metaphysician Undercover

and

(2)

They have no spatial-temporal separation, therefore no means for distinguishing one from the other, they are simply assumed to exist as a set. How do you think it is possible to order them when they have been conceived by denying all principles of order.? To introduce a principle of order would contradict the essential nature of these things. — Metaphysician Undercover

So there you are, still demanding that order must be temporal-spatial.

However, then, fifteen hours ago (not days ago), yes, you wrote:

Temporal/spatial was just one type of order, fishfry and Lluke gave examples of many other types. So we're not restricted to temporal/spatial order in our attempts at understanding the nature of inherent order. — Metaphysician Undercover

And that is the very remark that I just replied to. So I don't see you changing your mind since the post with (1) and (2) except the recent post of which I pointed out that it is inconsistent with your earlier stance.

/

And after so many days on end of you claiming that orderings are necessarily temporal-spatial, now you recognize that orderings do not have to be temporal-spatial, so what took you so long? It's piercingly clear that there are orderings that are not not temporal-spatial, but you could not see that because you are stubborn and obtuse.

Take K and make it an element of a set thus, {K} and what happens? It's, according to how it's defined, K again. That didn't quite go as planned, did it? — TheMadFool

No, it was as "planned", and consistent (without reguarity).

{{K}} = {K} = K which in plain English means K can't be made a member of another set. — TheMadFool

Wrong. That kind of nesting is consistent (without regularity). And it does not mean that K can't be a member of another set. I informed you about that multiple times. Proof:

Suppose K = {K}. Let ~x=K and ~xeK. Then ~ {x K} = K but K e {x K}.

What is wrong with you that you can't see that a set being a member of itself doesn't prohibit the set from being a member of other sets?

The whole exercise involving K = {K} is akin to claiming that b × 1 =/= b. — TheMadFool

That is argument by analogy, which is not valid for deduction such as mathematics. And the analogy even works against your claim.

Multiplication by 1 is idempotent.

With Z, the singleton operation is not idempotent. (So your analogy is backwards.)

With ZFC-R, it is consistent that there are sets for which the singleton operation is idempotent. (So your analogy DOES work in that context, and it works AGAINST your claim.)

So often, you are plainly illogical.

A set can't contain itself. Period! — TheMadFool

With Z set theory, there is no set that is a member of itself.

With ZFC-R, it is not inconsistent that there is a set that is a member of itself.

I gave you copious explanation about that. But you just go on your merry way ignoring the information you've been given.

Basically, it's just saying:

Suppose we have a set U whose members are all only those that are not the Russell set.

Let R be the subset of U such that R has every member of U except sets that are not members of themselves.

Some reasoning shows that R is not a member of itself, and that leads to some more reasoning that R is not a member of U. And some more reasoning leads to R is a member of itself if and only if R is not a member of itself.

So the initial supposition that there is a set whose members are all and only those that are not the Russel set is contradictory.

/

Basically, you wondered about avoiding Russell by a cutdown that deletes the Russell set from a universal set. So, instead of applying the Russell argument itself, I applied it to the cutdown, and still got the contradiction.

Somewhere you asked about having a set of all sets except the Russell set. I didn't send a proof that that doesn't work, but I want to now, because it's cute (basically reapplying the Russell argument.

1, EUAx(xeU <-> ~Az(zex <-> ~zez)) [premise]

i.e, assume, toward a contradiction, that there is a set of all sets except the Russell set.

2. Ax(xeU <-> ~Az(zex <-> ~zez)) [EG]

3. ERAy(yeR <-> (yeU & ~yey)) [separation]

i.e. take the subset of U that is the set of all sets in U that are not members of themselves.

4. Ay(yeR <-> (yeU & ~yey))

5. ReR <-> (ReU & ~ReR) [UI]

6. ReR [premise]

7. ~ReR [5, 6]

i.e. we have ~ReR since the premise ReR implies ~ReR.

8. ~ReU or ReR [5, 7]

9. ~ReU [7, 8]

10. ~ReU <-> Az(zeR <-> ~zez) [2 UI]

11. Az(zeR <-> ~zez) [9, 10]

12 ReR <-> ~ReR [UI]

13. ~EUAx(xeU <-> ~Az(xex <-> ~zez)) [12]

Read 'em and weep.

We would also have to look into the smallest unit of thought which I reckon is a single concept, each assigned an exclusive symbol of its own. — TheMadFool

Roughly speaking, those are the primitives.

it becomes more a linguistic issue than a logical one. — TheMadFool

It's a mathematical one.

Is the sense of this reproducible here, in a conversational way. in a non-onerous number of sentences? — tim wood

I can't do it some justice without some technicalities, but I will have to skip some defintions and to fudge some technicalities that would be handled better in a textbook. And to be cogent in a short space, I'll put some things in my own terms.

As is famous, Tarski proposed a correspondence notion of truth. For example:

'1+1 = 2' is true if and only if one plus one is two. [Using numerals and '+' and '=' on the left of the biconditional but words on the right of the biconditional, only to emphasize a certain difference explained in the next paragraph.]

That is not circular, since the '1+1 =2' is purely syntactical. and "'1+1 =2' is true" is a statement about the syntactical object '1+1=2', while the right side expresses a state of affairs.

Now, how do we formalize the notion of a 'state of affairs'?

Answer: With formal models.


A model is a certain kind of function from the signature (and also the universal quantifier in Enderton's book) of the formal object language:

The universal quantifier maps to a non-empty set called 'the universe'

n-ary predicate symbols (including n=0) map to n-ary relations on the universe.

n-ary function symbols (including n=0) map to n-ary functions on the universe.


For example, with the language of arithmetic, the standard model is:

the universal quantifier maps to the set of natural numbers

'=' maps to the identity relation on the set of natural numbers (that is "hardwired" since we are in a context of first order logic with identity)

'S' maps to the successor function on the set of natural numbers

'+' maps to the addition function on the set of natural numbers

'*' maps to the multiplication function on the set of natural numbers


Then the 'truth value' for sentences is inductively (mathematical induction, not empirical induction) defined (too many details for me to mention here). First are clauses for the denotations of the terms (atomic terms, then inductively, compound terms), then satisfaction for atomic formulas, then the connectives, then the quantifier, then a move from satisfaction of formulas to truth of sentences.

"Doesn't work" seems about as meta-mathematical as it can get. — tim wood

It's merely an informal heuristic expression.

Nowhere in the paper so far as I can understand it does he make clear either that or why it doesn't work with true. — tim wood

If that is the case (I don't recall all of that paper now), then it supports my point.

Anyway, his task was to prove the theorem, which he did. Explaining why it wouldn't work with 'true' is extra.

The proof works because 'provable' is arithmetizable while 'true' is not for a consistent theory. If 'true' were arithmetizable, then the theory would be inconsistent (Tarski).

True as not a formal concept? — tim wood

No. 'true' is formalized, though not in 1930. But the important thing for incompleteness is that 'true' is not arithmetizable in a consistent theory.

More exactly: a truth predicate cannot be defined in the language of a consistent theory. In other words, a predicate T such that Tn evaluates to true in the standard model if and only if n is the Godel-number of a sentence that evaluates to true in the standard model is not definable from the language of the theory. (I think I have that right.)

As far as I can tell, you don't understand the nature and motivation for formal languages. You would benefit from an introductory chapter in a book that explains it, and an introductory textbook on symbolic logic.

logically, these two arguments are identical, their premises are identical, their conclusions are too. — TheMadFool

A machine can check the second proof, But a machine cannot check the first proof (unless the machine had those English phrases programmed as fixed syntactical pieces, in which case the first argument would also be formal).

The way it seems to me, there's no point in talking about a book - the book being a message of some kind - in terms of how many words are in it. Similarly, proofs - logical entities - shouldn't be viewed as symbols. — TheMadFool

We are not saying that the import of a proof is the number of symbols in it.

(1) That translation is different from the one in the van Heijenoort book, which, if I recall correctly is the only one approved by Godel. I don't mention that to discredit your quote or the translation it came from. Rather, just to say that in general and in principle, it may be better to refer to the approved translation.

(2) Indeed, Godel mentioned that his proof deploys the liar paradox but with 'provable' instead of 'true'. But that is not itself the observation that if we substituted 'true' for 'provable' then the system is inconsistent.

(3) Godel may have made that observation (I don't recall), and it would seem obvious anyway, but it was Tarski who put the formal cherry on top with Tarski's theorem.

Why exactly he settled for provability over true would be interesting to know. — tim wood

That is clear from the proof. It doesn't work with 'true' but it works with 'provable'.

Did he recognize that truth is a metamathematical notion, not part of the mathematics itself? — tim wood

I'm not sure to how summarize Godel's view on that at the time of the proof. But that's not the reason for using provability rather than truth. The reason for using provability is that it works.

It's interesting that Godel landed on the idea of incompleteness from his failure to proof the consistency of analysis. Before incompleteness was even a twinkle in his eye, he was unsuccessfully trying to prove the consistency of analysis, and he saw an opportunity in that failure that would possibly prove incompleteness (I don't know the details about that though).

Note that subsequent to incompleteness, Tarski did provide a framework for handling 'truth' as a formal mathematical notion. It is metamathematical, but metamathematics is also mathematics. Mathematical logic and model theory are mathematics.

What I find problematic with defining proof length in terms of numbers of symbols in one is that it seems to miss the point — TheMadFool

No, because you don't understand the very particular point of this particular enquiry. This is not about saying how we should regard the notion of proof in some philosophical sense. Rather, it is a mathematical question about a "brute force" algorithm for generating formal proofs, which are sequences of sequences of symbols, and thereby enumerating the shortest proofs.

Whatever your views about what proofs should be, the context of mathematical logic (thus the context of the questions posed at the other forum) is that proofs are sequences of sequences of symbols (or other finitary characterizations such as certain kinds of trees, tableaux, et. al).

proof "lengths" must be measured, if possible, in terms of how many logical steps are taken from the start (premises) to the end (conclusion). — TheMadFool

That is one way to do it. We seem to get different answers to the "brute force" question depending on whether we ask about "the sum of the lengths of the formulas" or "the number of lines".

We are interested in how "hard" (scare quotes there are needed) it is a for a Turing machine to enumerate all the proofs. Toward that consideration, see that merely stating the number of lines might not be as instructive:

Consider a proof with only three lines, but the formulas on each of those lines are over a billion symbols each. That is a short proof in terms of lines, but not a short proof in terms of lengths of the expressions.

natural deduction seems to employ a classical method which consists of 3 propositions, 2 premises and 1 conclusion — TheMadFool

Where did you read such a thing?

In a conversational way, that's an okay summary. But it actually describes Tarski's result pursuant to incompleteness.

To me, it's odd that Church's theorem (undecidability of the set of theorems of the pure predicate calculus) and Tarski's theorem (the inexpressiblity of a truth predicate in a consistent theory) came in 1936, six years after incompleteness, but those two results are pretty easy corollaries. Why did it take six years to publish the proofs?

Hmm, maybe they depend on Rosser's 1936 improvement of Godel's result? I don't know.

One of these days I need to refresh my knowledge of the proof details for Church and Tarski results.

If you wish to view the law of identity as a "mystical" principle — Metaphysician Undercover

I don't.

If we assume that there is no [the inherent order], then we assume that the world is fundamentally unintelligible. — Metaphysician Undercover

Non sequitur.

So we don't know that there is an inherent order, we assume that there is, because that is the rational choice. — Metaphysician Undercover

I am sympathetic to the idea of assuming frameworks for making sense our experience. And, indeed, the mathematical notion of ordering can be part of that. It doesn't require an undefined notion of "THE inherent order".

I would consider that most good ontology is based in mysticism. — Metaphysician Undercover

To each his own.

See, not dogmatic. You expressed a philosophical preference, which is not itself an ignorant, confused and incorrect claim about mathematical logic and set theory, so I don't begrudge you having that preference.

Metaphysician Undercover fails to distinguish between two facts:

(1) For every set, there is a determined set of strict linear orderings of the set. (That is a simple and intelligible alternative to his confused notion of "THE inherent order" ensuing from identity.)

(2) For a finite set of cardinality n>1, there are n! strict linear orderings of the set. And there has not been given a definition of "THE inherent ordering" from among those n! orderings (expect Metaphysician Undercover's mumbo jumbo about an ordering based (based in what way?) on all relations among the members and such that no one can correctly identify it but that it exists by virtue of identity).

i cannot tell you the inherent order. It's not something that can be spoken,. — Metaphysician Undercover

Not only cannot you tell us what "THE inherent order" is for ANY set, but you can't even define the rubric.

You can't even say what is "THE inherent order" of {your-mouse your-keyboard} even though they are right in front of you. And you said a really crazy thing: If you identified "THE inherent order" of a set then we will identified it incorrectly. You have a 50% chance of correctly identifying "THE inherent order" of {your-mouse your-keyboard} but you say that you would be wrong no matter which order you identified as "THE inherent order"!

Temporal/spatial was just one type of order, fishfry and Lluke gave examples of many other types. So we're not restricted to temporal/spatial order in our attempts at understanding the nature of inherent order. — Metaphysician Undercover

It was pointed out to you that there are orderings that are not temporal-spatial. But you insisted, over and over, that temporal-spatial position is required for ordering.

But now you enlist the view of other posters, with whom you so strongly disagreed, to wiggle out of your own untenable view!

What has happened is that it has finally gotten through to you that ordering is NOT only temporal-spatial, so you shifted to saying that "THE inherent order" is based on "all the relations". A complete reversal of your position, except you still cling to your notion of "THE inherent order".

And you still can't say what "THE inherent order" is with regard to your new bases of "all relations".

truth of a determined order is dependent ONLY [bold and all-caps added] on our concepts of space and time. — Metaphysician Undercover

Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".

in the case of ordering, truth or falsity is dependent on the truth of our concepts of space and time [...] I think it can be demonstrated that each and every order imaginable is dependent on a spatial or temporal relation. To the right, left, or any such pattern, is spatial, and ANY INTELLIGIBLE SENSE OF "PRIOR" IS REDUCIBLE TO A TEMPORAL RELATION. I REALLY DO NOT THINK THERE IS ANY TYPE OF ORDER WHICH IS NOT BASED IN A SPATIAL OR TEMPORAL RELATION [bold and all-caps added]. — Metaphysician Undercover

Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".

If you truly think that there is some type of order which is intelligible without any spatial or temporal reference, you need to do a better job demonstrating and explaining it. — Metaphysician Undercover

Flat out contradicts your new claim; "we're not restricted to temporal/spatial order".

to grant to them "no inherent order" by denying them all spatial-temporal relations, you cannot now turn around and talk about their possible orders. ORDER IS A SPATIAL-TEMPORAL CONCEPT [bold and all-caps added] — Metaphysician Undercover

Surely, "first" does not mean "highest quality", or "best", in mathematics, so if it's not a temporal reference, what is it? — Metaphysician Undercover

You cannot remove the necessity of spatial-temporal positioning for those individuals, and still claim that the name refers to the individuals. — Metaphysician Undercover

It was my suggestion that "order" is fundamentally temporal — Metaphysician Undercover

If it can't be understood without spatial or temporal reference, then there clearly is a need for space and time in math, or else all mathematics would be simply unintelligible. — Metaphysician Undercover

And understanding them is what requires spatial and temporal reference. The number 5 has no meaning, and cannot be understood without such reference. — Metaphysician Undercover

the ordering of numbers requires a spatial or temporal reference. — Metaphysician Undercover

If mathematics talks about an order which is not temporally, nor spatially grounded, then I think such a mathematics would be nonsensical. I've seen some people argue for a "logical order" which is neither temporal nor spatial, but this so-called logical order, which is usually expressed in terms of first and second, is always reducible to a temporal order. — Metaphysician Undercover

What is this act which you call "the collecting of the objects into a set"? Wouldn't such an act necessarily create an order, if only just a temporal order according to which ones are collected first? — Metaphysician Undercover

I have opted for a sort of compromise to this problem of justifying the pure a priori, by concluding that time itself is non-empirical, thus justifying the temporal order of first, second, third, etc., as purely a priori. — Metaphysician Undercover

If you want to define numbers by order, then you assign temporality as the difference between 1 ,2,3 and 4. — Metaphysician Undercover

You are arguing a philosophy of truth. — Metaphysician Undercover

I made no argument for a philosophy regarding truth.

empirical validation isn't relevant.
— Wayfarer

That's exactly what makes arguing for mathematics as the purveyor of truth, dogmatism. — Metaphysician Undercover

And I didn't argue that mathematics is a purveyor truth known a priori and that empirical concerns are not relevant.

Your claim that I am dogmatic is unsupported.

if I stated an order, it would be a representation, imposed from my perspective, and therefore not the order which inheres within the object, the inherent order. — Metaphysician Undercover

And you chided ME for my interest in mathematics that you deem not empirically justified. Now you're resorting to a mystical "true order that inheres in an object" but such that if we even attempted to state what that order would be, then we would be wrong!

Just by saying which you think is "THE inherent order" you would necessarily not be choosing "THE inherent order"? So just by saying which ordering you think is "THE inherent order" you would necessarily be wrong!?

As I explained, the objects, as existing objects, have an inherent order — Metaphysician Undercover

As you dogmatically claim. You keep skipping the central challenge to your claim. That challenge will be repeated in this post.

The inherent order is the true order, which inheres in the arrangement of objects. — Metaphysician Undercover

Petitio principii!

You just shift the answer among various still undefined terms: "actual", "inherent", and now "true".

What is "THE true order" such that the other orders are not "true" orders?

The inherent order, if we were to attempt to describe it, would contain all the truthful relations between those beings — Metaphysician Undercover

Before, it was temporal/spatial. But now, for just three people it's an even more complicated big deal with you just to attempt to describe "THE inherent order". And "all the truthful relations". I guess you mean the various comparisons, connections, associations, differences, shared properties, contrary properties, contexts, etc. among those three people. That could be a vast number of things. And still you haven't defined how "THE inherent order" is determined by a vast number of associated properties among the three people.

Here is a set of three living people you know about:

{Angela Merkel, Lance Armstrong, Justin Bieber}

Now, please tell me "THE inherent order" of them. Please tell me how you used "all the truthful relations" to determine "THE inherent order".

The inherent order is the true order, which inheres in the arrangement of objects. — Metaphysician Undercover

"The true order that inheres", but you can't say what it is. Sounds pretty "abstract", nay mystical, to me.

Order is the condition under which every part is in its right place. — Metaphysician Undercover

From "inherent" to "actual" to "true" to "in its right place". None of them defined by you.

'is prime' is a predicate, not an ordering.
— TonesInDeepFreeze

It is a predicate which refers to relations with others, therefore an order. — Metaphysician Undercover

So any predicate that involves "relations with others" is an order?

'the queen of hearts is a red card' and 'the four of clubs is a black card' are statements about predicates of the two cards. So what is "THE inherent order" - <QH 4C> or <4C QH>? Do we also have to consider all the "truthful relations between them"? What are some of those "truthful relations"? How did you determine "The inherent order" based on those "truthful relations"?

By the way, what do you mean by 'truthful' that wouldn't be said by just 'true'? And why even say 'truthful relations' when you could just say 'relations'?

according to fishfry ordering was removed, abstracted away, — Metaphysician Undercover

I don't speak for him, but I would imagine he's using such locutions as figures of speech.

I'll say it for you again, without recourse to figures of speech such as "removing" as I did before without recourse to figures of speech such as "removing":

{Bob Sue Tom} is the set whose members are all and only Bob, Sue and Tom.

There are 6 strict linear orderings of the set {Bob Sue Tom}.

If you want to say there is one of those orderings that is "THE inherent/actual/true/truthful/tutti-fruiti" ordering while the others are not "THE inherent/actual/true/truthful/tutti-fruiti" ordering, then you need to DEFINE the terminology "THE inherent/actual/true/truthful/tutti-fruiti" ordering".

If you are having difficulty with "inherent order", it is fishfry's term as well — Metaphysician Undercover

Again, I don't speak for him, but I understood him to be making a correct point that does not depend on whether he used a figure of speech such as "inherent". His point was that the set is determined by its members and not by the orderings of the set. I don't speak for him, but I have little doubt that he would agree (without using "inherent"):

(1) A set is determined by its members. Axiomatically, S =T if and only if every member of S is a member of T and every member of T is a member of S.

(2) A finite set of cardinality n has n! strict linear orderings. So for sets with cardinality greater than 1, there are more than 1 strict linear orderings of the set.

See, no need to use the word 'inherent' and especially not "THE inherent ordering".

Saying "the set has no inherent ordering" boils down as a locution to saying (1) and (2).

But then YOU jumped to say there IS "THE inherent ordering", and you are floundering to meet the challenge of defining it.

He doesn't have the actual superiority to spend. He's in overdraft with just that one pathetic attempt.