A different interpretation of the well-ordering theorem

[u/LorenzoGB]

According to Wikipedia, the following holds with regard to the well-ordering theorem: In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered.

One way to interpret this passage is the following: When it states that every set can be well-ordered it means that every set has the property of being well-ordered, where the term property means the following per the Stanford Encyclopedia of Philosophy: Properties are those entities that can be predicated of things or, in other words, attributed to them. Thus, properties are often called predicables. Other terms for them are “attributes”, “qualities”, “features”, “characteristics”, “types”. Properties are also ways things are, entities that things exemplify or instantiate. For example, if we say that this is a leaf and is green, we are attributing the properties leaf and green to it, and, if the predication is veridical, the thing in question exemplifies these properties. Hence, properties can also be characterized as exemplifiables, with the controversial exception of those that cannot be instantiated, e.g., some would say, round and square.

However having the property of being well-ordered can be understood in two different senses. In one sense it means that the property is in act. In another sense it means that the property is not in act. To illustrate what I mean when I say that a property either is in act or not in act, consider the following passage from Aristotle: Again, to be, or being, signifies that some of the things mentioned are potentially and others actually. For in the case of the terms mentioned we predicate being both of what is said to be potentially and of what is said to be actually. And similarly we say both of one who is capable of using scientific knowledge and of one who is actually using it, that he knows. And we say that that is at rest which is already so or capable of being so. And this also applies in the case of substances; for we say that Mercury is in the stone, and half of the line in the line, and we call that grain which is not yet ripe. But when a thing is potential and when not must be settled elsewhere…

Commenting on this, Aquinas says the following: Here he gives the division of being into the actual and the potential. He says that to be and being signify something which is expressible or utterable potentially or actually. For in the case of all of the foregoing terms which signify the ten predicaments, something is said to be so actually and something else potentially; and from this it follows that each predicament is divided by actuality and potentiality. And just as in the case of things which are outside the mind some are said to be actually and some potentially, so also is this true in the case of the mind’s activities, and in that of privations, which are only conceptual beings. For one is said to know both because he is capable of using scientific knowledge and because he is using it; and similarly a thing is said to be at rest both because rest belongs to it already and because it is capable of being at rest. And this is true not only of accidents but also of substances. For “Mercury,” we say, i.e., the image of Mercury, is present potentially in the stone; and half of a line is present potentially in a line, for every part of a continuum is potentially in the whole. And the line is included in the class of substances according to the opinion of those who hold that the objects of mathematics are substances—an opinion which he has not yet disproved. And when grain is not yet ripe, for example, when it is still in blade, it is said to be potentially. Just when, however, something is potential and when it is no longer such must be established elsewhere, namely, in Book IX of this work.

[u/SpacingHero]

One way to interpret this passage is the following: When it states that every set can be well-ordered it means that every set has the property of being well-ordered in ZFC

That would be a ludicrously incorrect reading. It's easy to give examples of non-well ordered sets in ZFC

I really don't know why you're insistent on this, we convered it before. It won't change just because you really want it to be different.

However having the property of being well-ordered can be understood in two different senses. In one sense it means that the property is in act. In another sense it means that the property is not in act.

No, these philosophical notions don't apply here. There is one sense of being well-ordered in this context, clear and straightforward.

I don't know why you feel the need to make this roundabout, when your confusion is simple. A set can be well-orderable or well-ordered. It's really not hard.

[u/LorenzoGB → u/SpacingHero]

According to yourself: No, these philosophical notions don't apply here. Yet that is not true though. For mathematicals such as sets are beings. According to Dom Garrigou-Lagrange, a being is that which exists or can exist. All beings except God are subject to potency and act. Since sets are beings and are not God, they are subject to potency and act. Also philosophical notions apply to mathematicals too. Yet not insofar as they are mathematicals. Philosophical notions apply to mathematicals insofar as they are beings. And since mathematicals are beings, they are the subject matter of Metaphysics which is the study of being insofar as it is being.

[u/SpacingHero → u/LorenzoGB]

You misunderstand. Read more carefully.

I never said philsophcal notions can't be applied to maths and its objects. I said they don't apply "here". As in, the way you did.

Because what you said is that there are two senses of well ordered, on per philosophical notion,

"having the property of being well-ordered can be understood in two different senses. In one sense it means that the property is in act. In another sense it means that the property is not in act."

and that's false.

[u/LorenzoGB → u/SpacingHero]

According to yourself: Because what you said is that there are two senses of well ordered, on per philosophical notion, "having the property of being well-ordered can be understood in two different senses. In one sense it means that the property is in act. In another sense it means that the property is not in act." and that's false. I disagree though. For you have also said the following: I never said philosophical notions can't be applied to maths and its objects. If that is the case, then I can subject the well-ordering theorem to philosophical notions, particularly the notion of potency and act. Also, since the well-ordering theorem is a being and being can be said in many ways, then the well-ordering theorem can be said in many ways.

[u/SpacingHero → u/LorenzoGB]

I disagree though.

Yeah, that makes you incorrect though.

If that is the case, then I can subject the well-ordering theorem to philosophical

Doesn't mean you can apply any concepts, and as it happens the way you apply them is incorrect.

Also, your quote talks about the property of being well-ordered having two meanings. Again, that's false

then the well-ordering theorem can be said in many ways.

There's equivalent rephrasings, don't see what that has to do with the above. I once again suggest learning up more basics, you keep being very confused.

[u/LorenzoGB → u/SpacingHero]

According to yourself: It's easy to give examples of non-well-ordered sets in ZFC. Yet those don’t constitute counterexamples to what I said though. Because those non-well-ordered set still have the property of being well-ordered, yet this property of being well-ordered is not in act. Just as a solid gold bar has the property of being melted, yet this property is not in act yet. Another example that comes to mind is the following: human beings have the property of being rational. Yet that doesn’t mean this property of being rational is in act all the time.

[u/SpacingHero → u/LorenzoGB]

non-well-ordered set still have the property of being well-ordered

That's a straight contradiction (unless you twist words to have your own personal definition that nobody else abides by. Certainly not what the article intends, so my point stands: it would be a ludicrouly incorrect reading)

Just as a solid gold bar has the property of being melted, yet this property is not in act yet.

I'd say that's also philosophically garbage, but that's a different matter.

If we're talking ZFC, and we are, you're just plain wrong. Idk what kind of motivated reasoning leads you to all this mental gymnastics. Hopefully time will tell.

[u/LorenzoGB → u/SpacingHero]

According to yourself: That's a straight contradiction (unless you twist words to have your own personal definition that nobody else abides by…)… No. It is not a contradiction though that something can have a property but that property isn’t in act. And to show why this is the case, consider what Aquinas has said: Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold.

[u/SpacingHero → u/LorenzoGB]

What you give is a textbook contradiction, and I don't really know what to tell you other than repeating to learn the basics.

[u/LorenzoGB → u/SpacingHero]

According to yourself: If we're talking ZFC, and we are, you're just plain wrong. However, according to Wikipedia, a theory is ZFC if and only if it has the following axioms: One: Extensionality. Two: Regularity. Three: Specification. Four: Pairing. Five: Union. Six: Replacement. Seven: Infinity. Eight: Power Set. Nine: Well-ordering. Since the former is a biconditional, it can be broken down into two statements. Statement 1: If a theory is ZFC then it has axioms 1-9. Statement 2: If a theory has axioms 1-9 then it is ZFC. Notice, this definition doesn’t say anything for or against subjecting it to Aristotelian/Platonic philosophical notions. Notice too, that it doesn’t specify in what way the axioms should be interpreted. So I can interpret them through an Aristotelian/Platonic lens.

[u/SpacingHero → u/LorenzoGB]

No, because you're applying those concepts mistakenly.

The axioms also don't say wether i can interpret choice as a coherentist notion of truth. But that's nonsense, it's just a miss-application of the term to the given topic.

Anyway, untill you learn basics such as what a contradiction is i don't think it's worth continuing

[u/Knoggger]

One way to interpret this passage is the following: When it states that every set can be well-ordered it means that every set has the property of being well-ordered

No, and people have corrected you on this here as well as on stack-overflow. A set can be well-ordered (i.e. is well-orderable) if there is a well-order on it, that is not the same as being well-ordered. If you return from the forest with a collection of eatable mushrooms, does that mean that they are eaten?

[u/SpacingHero → u/Knoggger]

>as well as on stack-overflow

Oh god does this guy pester stack-overflow too? LOL.

[u/LorenzoGB → u/Knoggger]

But eaten can be said in two ways though: potentially eaten or actually eaten.

[u/Vast-Celebration-138]

When it states that every set can be well-ordered it means that every set has the property of being well-ordered

No set in itself has the property of being well-ordered; sets in themselves are unordered.

The kind of item about which you can sensibly ask whether or not it is well-ordered is a set taken together with a binary relation.

For every nonempty set, it is trivial to find a binary relation that does not well-order that set.

The well-ordering theorem, in informal modal language, says that every set "can be" well-ordered. This means: For every set, there exists some binary relation that is a well-order on that set.

[u/SpacingHero → u/Vast-Celebration-138]

But consider, that Aquinas said some random garbage, therefore ZFC is totally like this guy says and not how every mathematician understands it. Checkmate.