As it turns out, the mixing of the concepts of naturalism and materialism (that these terms are intended to be interchangeable) does show that you've made the idea of God impossible. The concept of God that we have in this discussion is (implicitly) that God is immaterial, which is a direct contradiction to materialism.

So while you may not have stated that explicitly, it's implicitly true based on your use of language.

If you are claiming a "scientific paradigm" then by your own concepts you have precluded the existence of God.

You ask some excellent questions!

Since they are basically questions that Philosophers grapple with , I would be delighted to discuss them with you in the SMP Forum.

Start a thread there, and we can talk about those important subjects.

... I don't want to take the time to unravel the last four posts of failures to understand a really simple argument and to constantly correct your misrepresentations of my position or your claims that I've not answered questions that I've done my best, in good faith, to answer over and over. It's too tiring to try to pick out the bits that matter in amongst all the personal attacks and projections.

I think we're done here Aaron, this is serving no useful purpose for me.

I'm not going to block you, you might still say something useful. So have at it.

Assuming you're not winding me up, I might do that, I find Epistemology fascinating.

It's true that you often don't want to make the effort to extend your knowledge. And this is one of your major shortcomings in these conversations. You don't take the time to understand what you're objecting to, and so your objections end up being meaningless.

But because you're not sufficiently humble in your intellect, you just assume you know what you're saying even though it's clear that you don't. You assume your statements are already correct, and look for ways to justify their correctness, rather than wondering if your statement may be wrong.

Your ability to manage a relatively simple statement about math (determining whether it is a necessary or contingent truth) is a clear example of this.

But you're free to carry along in ignorance. That's what you've done for years, and that's what one would expect that you will continue to do.

Kewl! See you there!

I don't know your math background, but I've already shown a model L where 1+1>2 in my post. The axiomatic justification for the regular integers (where the supposed necessary truth that 1+1=2 holds) is consistent if and only if my L integers are consistent as well. So it seems pretty trivial to say arithmetical facts are contingent on which model of the integers you are talking about thanks to the fact that consistent axiomatic formulations of arithmetic are not allowed to prove their own consistency. If you want to say my system is not sound, I can just simply assert the same thing about yours. It goes back to my original point of logic being a poor tool to differentiate between basic true and false, much less a new animal like necessary truth.

Look kids, this is called 'projection"...

(I believe PA is undecidably consistent or inconsistent. So I'm using that as the basis of what follows.) If PA is consistent, then 1. and 2. together is a logical contradiction. Also, if PA is inconsistent, then 2. doesn't add anything.

So I'm not entirely sure what you're saying when you claim that L and PA are either both consistent or both inconsistent.

I have a PhD in math (analytic number theory), but have spent very little time doing formal logic. I'm vaguely familiar with the history of mathematics tracking through the mathematical formalism of the late 1800s and things like Godel's incompleteness theorems, but I don't claim to have a deep perspective in those mathematical foundations.

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The actual choice of "1+1=2" is just a stand-in for any statement that you might determine is logically necessary. It can be replaced with a syllogism such as

P1) Socrates is a man
P2) All men are mortal
C) Socrates is mortal

Soundness has little to with the discussion. You would have to take on a tangential problem of what is meant by math being "true." For example, you could take the stance of a mathematical fictionalist and claim that mathematical objects don't even exist, and math is only true because it's vacuously true. A nominalist doesn't need to declare a position either way.

You would still need to settle in on a concept of what you think a "possible universe" would be. If you think that every set of non-contradictory mathematical axioms is itself a universe, then you would be correct in your understanding of how you've established the terms.

However, that framework is not particularly consistent with the usage within philosophy, at least as far as I understand it. There are multiple philosophical concepts of "possible worlds" (or possible universes):

https://plato.stanford.edu/entries/possible-worlds/

No. If PA is inconsistent it can be shown in PA. If PA is consistent, it cannot be shown in PA, but it can be shown in ZF or other systems.

Bolded is wrong by Godel's 2nd incompleteness theorem. That and PA and L are co-consistent is a basic theorem from a logic 101 class, so I don't see much point in arguing about it. You can take my word for it or not.

Arithmetical statements can be expressed in PA so they are subject to the type of model building I am engaging in because Godel's completeness theorem applies. That's not the case for things like basic syllogisms.

Seems like soundness has everything to do with the discussion since you can't reject L on the basis of consistency. Soundness gives a perfectly valid reason to reject L. 1+1 does actually equal 2 . But you're left with the fact that it's just a "contingent" truth, based on the fact that we are talking about the normal integers and not L. I think that is a reasonable (standard?) view.

I pretty much think this is solved. Arithmetic truth cannot be formalized in arithmetic. The rest doesn't matter imo.

I am only saying it's crazy to claim both that math exists/is non-material AND arithmetical facts are necessary truths when it's so easy to construct non standard models that "exist" and contradict facts you claim are "necessary" truths. Rejecting either or both is fine.

I don't know what these words mean in philosophy either or whether their usage makes sense in whatever context they use. I do think the usage itt is nonsensical wrt arithmetic though.

Okay. But that doesn't answer the question of decidabiity.

It seems you're misstating the ideas. Godel's 2nd theorem is about the ability to prove consistency, not logical consistency in and of itself.

I think you're wandering off to some strange areas where it's not clear that what you're saying follows.

Godel 1 has to do with unprovable statements. Basically, there exist statements that cannot be proven either true or false.

Godel 2 has to do with a system being unable to prove itself consistent.

Neither of these has to do with "truth." Unless the only things that can be considered true are things that can be proven mathematically true. But that's a very heavy statement.

If you are approaching things from a purely formal view of the universe, I think you would be correct.

But I don't know that this is true under all formulations of the ontology of mathematics. For example, nominalism uses physical reality as a representational foundation of mathematics. "1+1=2" is true because it is verified by every physical manifestation of it. (If I take a collection containing one object, another collection containing one object, and put them together, then I get two objects. This is, of course, subject to having a meaningful concept of an object, and subject to the objects being able to be "combined" in a way reflects the concept of addition, so that a proton and anti-proton being "added together" is not represented by "1+1.")

From a Platonic point of view, I think this runs into problems. "1" is a specific object that has specific properties. Addition of those objects is something that's well-defined conceptually.

Okay. Nobody has claimed that arithmetic is being formalized "in" arithmetic.

Do you know what decidable means in math? I think it does answer the question of decidability of Con(PA) as well possible. If you want to talk about the truth of Con(PA) that's something else. To restate, If PA is inconsistent, it is decidable in PA. In that case PA must prove itself inconsistent. If it's consistent you have to go outside of PA to prove Con(PA). How else can you answer the question of decidability?
The next portion will maybe clear up the confusion about the above.

I can assure you I'm not. I have a Phd in math and this is undergrad level stuff if you have doubts. Here is a popular level statement of Godel's 2nd incompleteness theorem

The bold is what guarantees the consistency of L. I couldn't find a good link to a logic 101 lecture in 5 min, but the PA + Not Con(PA) case is usually presented as an example.

Stating it again, If PA is consistent, PA + Not Con(PA) must be consistent. That's bc if PA is consistent and PA + Not Con(PA) is inconsistent, PA would prove Con(PA), which contradicts the bolded above. I can't really respond to the rest of your post bc I don't think you understood the post you were responding to.

So I started a thread but I decided to ask about moral theories since the philosophy of morality is also a subject of great interest to me.

I discovered an interest because I was posting here and seeing the question of atheism and morality coming up over and over. In fact, the guy I quoted in my OP I this thread doesn't think that it's possible for atheists to have morals. Naturally, I disagree.

Perhaps what he means is that he doesn't think that we can really have moral values, we can only just make things up to suit ourselves. I wouldn't disagree that that's what I'm doing with two caveats.. 1) I'm not simply making it up, I have done some reasoning on the issue with guidance from what I've learned about the philosophy of morality, 2) I think that's what everyone is doing since I have no reason to believe that there are gods for objective moral values to come from.

I gave my OP guy the Euthyphro dilemma and surprisingly he wasn't able to counter it, in fact he called it a 'terrible argument'... which leads me to believe that he didn't recognise it, it's quite something to call an argument by Plato 'terrible'... plus he's from Reasonable Faith and Craig has countered the dilemma, so why not give me Craig's answer? Who knows..

I will "chime in" in your new thread.

The ability to prove or disprove a statement within a particular set of assumptions. Euclid's 5th axiom is undecidable in the sense that it cannot be proven either true or false given the other four axioms.

Thanks for clarifying that. I did get myself confused trying chase down two different trains of thought at the same time and not being careful enough.

In terms of L:

Here's what I said:

There's a collision of concepts there.

I stand by my claim of logical contradiction: If P is any statement, then P and not(P) is a logical contradiction. Thus Con(PA) and not(Con(PA)) would be a logical contradiction.

However, mathematical consistency is about the ability to generate statements. If S is the set of all valid statements that can be formulated within a particular mathematical theory (collection of axioms) then that theory is consistent if there exist no statements P for which both P and not P are in S.

You're saying that if PA is consistent, then you can never derive a statement that contradicts 2. And it is within that framework that you disagree with the idea that there is a contradiction.

So there's a gap between our concepts of what it means to be a "contradiction." I would say that I'm using "(logical) contradiction" as a statement about truth values, whereas you're using "(provable) contradiction" as a statement of statements that can be proven within a particular mathematical theory. And that distinction is important.

I also want to go back to the core statement:

It is an open question as to whether the ability to assert a "consistent" set of axioms is what it means to be a "possible world." For example Euclidean and hyperbolic geometry are both logically consistent in "this world." But at most, only one of them is true (and it's possible that neither is true).

So, returning to the sentence that followed:

It's not clear that the ability to assert a consistent axiomatic framework is enough to assert a "possible" world. There's much work to be done to establish what makes a possible world possible.

This is where I started into mathematical fictionalism and nominalism. Because while "arithmetical facts" (as understood as the consequence of particular choices of axioms) may depend on the selected model that is chosen, you have to declare a status of those facts as facts of the universe. I can simultaneously hold that "the Pythagorean Theorem is true" (in the context of Euclidean geometry) and that "the Pythagorean Theorem is false" (in the context of hyperbolic geometry) while existing in this universe. And none of this changes if it turns out that "the Pythagorean Theorem is false" as a statement about the universe itself. Facts of arithmetic are not necessarily facts of the universe, and facts of some axiomatic formulations are certainly NOT facts of the universe.

So the mere assertion of axioms that imply "1+1=2" is false does not immediately imply that there is a "possible universe" in which "1+1=2" is false. But you're right that there is space to consider that, but it would require elaboration on the idea of what it actually would mean to have a "possible universe."

(Note: I swapped between "possible world" and "possible universe." I think "possible world" is a more philosophy-ish phrasing and "possible universe" is more of a physics-ish phrasing.)

You are making a substitution error here. This is not the same as L. The above is a "logical contradiction", but PA and Con(PA) are two different things. I am talking about PA+not(Con(PA)) which is consistent and completely different from Con(PA) + not(Con(PA)), which is inconsistent.

Another way to see this is to think about PA +Con(PA) which is obviously not the same as PA. Specifically, Godel sentences in PA, which PA cannot prove by Godel's 1st are provable in PA +Con(PA). I think if you get this you'll see I'm not using contradiction, consistent or any of these words in any manner other than what they must mean. Maybe the rest of my posts will make sense after you understand this?

Right. But then this is the statement you bolded and objected to.

You are using "consistent" to mean "unable to prove a contradiction." I used "consistent" to mean "the non-existence of a contradiction." If you assume PA (1) and it turns out that PA is consistent, then adding in that PA is inconsistent (2) creates a contradiction. Godel 2 claims that if you assume PA (1) and it turns out that PA is consistent, then you will never generate a statement that contradicts (2).

I would definitely challenge that there is a singular interpretation of these words because I'm not limiting limiting statements to just provable statements and yet you are.

So you think PA + Not Con(PA) is inconsistent if PA is consistent? There is just no interpretation of consistent in math or philosophy even where that is true. If you want to bet I could probably find a source pretty easily that says the opposite.

Maybe this mathy stuff should be moved to SMP?

There isn't much math in SMP either these days. I don't think we he have much else to discuss though so no need to move.

Actually, whenever you see "PA", and think "Probable Ass*ole, it makes it quite funny.

I think PA + not(Con(PA)) is a logical contradiction if "PA is consistent" is true. The reason for this is that your system would have both the statement Con(PA) and not(Con(PA)) being true.

I welcome you to find that source. I've already laid out the terms and have identified the distinction and shown you the meaning of the terms in a way that separates out what you're saying from what I'm saying.

I'll try it like this: A set of statements S is consistent if S does not a statement P for which not(P) is also contained in S.

In our considerations, we are using different sets S. You seem to be using the set of provable statements. I'm using the set of true statements. You're declaring that "the set of true statements is identical to the set of provable statements." I reject that. I hold open the existence of true statements that cannot be proven.

LOL -- I've never seen anyone use PA like that before.

My system does NOT have the statement Con(PA) being true anywhere in it. Godel's 2nd specifically tells us that it cannot.

I don't know why you are switching to "logical contradiction" instead of just inconsistent.

If PA is consistent, do you think ~Con(PA) by itself is a logical contradiction? That's really the only way to save what you said, but it really tortures the meaning of contradiction into just a synonym for false.

The error your making is confusing basic consistency with soundness or intuitively true or a higher level concept like w consistency.

Here

I'm simply using that fact, plus the Completeness theorem, which guarantees a model for L. That's really all I'm saying and you trying to come up with crazy new definitions for contradiction or consistent doesn't prevent me from creating my model.

Of course I never said that. Not even sure why you think I did tbh.

I'm shocked.