zurew

You are awake, when you start to talk informatively like :

Here is a question to the math and logic folks: Im not sure whether the following conclusion necessarily/inevitably/deductively follows from the second incompletness theorem or not, because I have seen that theorem phrased in different ways and it is a little bit confusing to me how to apply negation to it properly. Because I understand, that under classical logic , its easy to just apply negation to a statement (if it is phrased in a clear way) and get what deductively follows from it. So - When you have a sufficiently complex system that can prove its own consistency, does that system necessarily have to entail contradiction or not? Some things that I ve gathered from the second incompleteness theorem so far: 1) there are sufficiently complex systems that cannot prove their own consistency, but just because they cannot prove their own consistency doesn't deductively follow that they are inconsistent (I have shown examples of this in my previous posts, where a system's consistency was shown by another system). 2) Sufficiently complex Inconsistent systems can paradoxically prove their own consistency, but this is the part im unsure about - just because sufficiently complex inconsistent systems can prove their own consistency, doesn't necessarily mean that only inconsistent systems are capable of proving their consistency (Only inconsistent system being capable of proving their consistency maybe deducitvely follows from the second incompleteness theorem, but im not sure of it yet, this is why im asking whether it does or doesn't follow from it deductively) Leo btw this is why I was pushing back on this specific thing, because it was unclear to me whether this: "sufficiently complex system that can prove their own consistency, necessarily have to be inconsistent or in other words necessarily have to contain a contradiction" deductively follows from the second theorem or not, because if it does, then I will obviously concede this to you (but it isn't obvious to me whether it is inductively or deductively follows from it , this is why I was asking you im my previous post to show how it follows deductively) If its unclear what Im asking I will try to rephrase it.

Im not sure I undestand exactly what you mean by absolute consistency. Do you mean something like a system is absolutely consistent if it can prove its own consistency within itself (without any need for an outside system to prove its consistency)? and by relative consistency I assume you mean a system that cant prove its own consistency within itself, but its consistency can be proven by another system.

Thats an interesting cheap shot from someone, who still doesn't have an understanding of prop logic and what a valid inference is (it would literally take you roughly 2-3 days to learn it , but you still refuse to do it). Being a critical thinker doesn't mean making logical jumps and just freely inferring whatever you feel like will confirm your underlying biases - it means rigorously and extremely carefully investigating the logical entailments and implications of a statement or of a conclusion - regardless if that confirms or undermines your existing biases or beliefs. Speaking of confirming ones underlying biases Leo, you are either intellectually dishonest (you are not honest about the limits of your argument - meaning you know that the philosophical implication you are talking about is not based on a deductive inference or in other words, the conclusion of your inference doesn't necessarily or inevitably follows from the incompleteness theorems) or you don't have an understanding of whats the difference between a deductive an inductive and an abductive inference and because of it, you think your inference is more powerful than it actually is. You should learn from Gödel , because he was actually intellectually honest about the limitations regarding the inference that he made about his own theorems' philosophical implications ( he didn't pretend, that his conclusion about philosophy deductively follows from the incompleteness theorems): As a sidenote: The philosophical implications are still very much argued and nothing is settled on this as you originally tried to imply. Now I will somewhat address your statements about "technical details" and how you labeling yourself as a big picture thinker won't get you out of the trouble and won't excuse your mistakes: There is this weird idea in your mind, that if you are a big picture thinker, you don't need to apply any rigor in your thinking anymore and or that you can just get away with making loose inferences. You and some of the people on this forum mistake rigor to mean something like - " focusing on all the irrelevant tiny details, regardless if that has almost no effect on the big picture or on the given proposition that we disagree on ", but thats just wrong and false. Being rigorous means indentifying all the relevant details (relevant there means: details that could change the truth value of a given proposition) and then carefully going through each and every one of those relevant details and investigating the logical relationships between those details. You can make valid and invalid inferences and If you care about making valid inferences (valid there means that it is impossible for the premises to be true and the conclusion to be false) and if you care about reaching valid conclusions about big picture propositions - you better apply as much rigor to your thinking as possible. If you are not rigorous, you will mistake an invalid inference for a valid inference ( invalid inference - where the conclusion doesn't inevitably/necessarily follows from the premises - in other words, invalid inference is when there are logically possible scenarios where the premises that you infer from are true and your conclusion is false) Or if you don't want to hold yourself to such a standard, where you only let yourself to exclusively make deductive inferences, then you should be very honest and aware about the limitations of your non-deductive inferences. Now to clear up why I asked for a source: The reason why I asked for citation or for a reference is because you have made some claims and then tried to use the incompleteness theorems as a justification for some of those claims . We have to be clear and careful - there is a giant difference between a source explicitly confirming something and between you making a loose inference to confirm your bias using a premise from the source (unless it is a deductive inference) . Now, in this specific case, the actual reason why thats problematic, is because the incompleteness theorems literally contradict (its not that they don't explicitly state some of your conclusions , and that your conclusions in principle could be inferred from the incompleteness theorems, but they literally contradict) some of your claims. So one claim that they contradict is this: Both the Peano arithmetic and ZFC are considered to be very powerful formal systems or in your words: "they are capable grasping a significant amount of reality" and none of them are inconsistent/contain a contradiction as you assert that they should inevitably become inconsistent. So both of those examples directly contradicts your claim that "If your formal system actually tried fo grasp any significant amount of reality it would contradict itself." Now, it is possible that you mean something different compared to what mathematicians mean by a 'powerful enough' formal/axiomatic system ), but then again, you come back to the issue of this: You cannot pretend that these theorems are directly justifying your philosophical biases unless, you can spell out a deductive inference (which I very highly doubt you can) that ends with whatever philosophical conclusion you want to get out of this. Just as a sidenote , to me this just seems like a desperate attempt on your part to use the incompleteness theorems to try to justify some of your philosophical biases and thats why you don't want to admit that you are wrong and that you have made some incorrect statements. If your conclusions from this would deductively follow I would have 0 issue with it or if you would admit that your conclusion does not necessarily/inevitably follow I would also have 0 issue with that. Just be intellectually honest and be rigorous. Regarding the self-reference statements, I would be curious what would be the issue with these: 1) Set of all sets (S), where S contains itself 2) "This statement is true" If there isn't any issue with the two above, then we have to sleep with the conclusion that not all self-referencing statements will inevitably lead to a paradox, just only a subset of self-referencing statements.

@Leo Gura Whatever you are referencing there cannot be found in any of the two Gödel incompleteness theorems ( in fact, I already showed you a counterexample in my previous post). I linked you two separate links there (one is wiki and the other one is SEP) and highlighted the main things. Now, its possible that you are talking about something that im not familiar with, but Im confident that whatever inconsistency you are talking about there, cannot be found in any of the 2 Gödel's incompleteness theorems and if you think it can be found there - show me where, please. Or if you want to change your position to "yes, okay, I was wrong, it cant be found in none of the 2 Gödel's incompleteness theorems" Im okay with that also, just show me something that substantiates this claim "Self-reference will always make your system inconsistent."

I would slightly disagree with this and would say that he isn't just making a slight language mistake there (like I am not getting hung up on little semantics), but his main point is basically wrong . (claiming that something is inconsistent is substantively different from the claim that a system cannot prove its own consistency within itself) - those are two completely separate claims and not even remotely similar in meaning and in implications (imo). Okay, I won't drag you into this debate with Leo , i was just checking whether you had any disagreement with me since you have a much higher understanding of math than me and you have been making substantive points throughout the whole thread.

@Ero Do you have a disagreement with any of this? (feel free to call out the bullshit, don't hold back) Or if you prefer wikipedia:

Lol this thread has become a high school (and at certain periods an elementary school) math boot camp. @Nemra I don't know how you have this much patience - hats off to you.

@Leo GuraSo in other words, from system's inability to prove its own consistency doesn't necessarily follow, that that system is actually inconsistent ( at least thats my understanding). Or if you prefer wikipedia: Leo, Im sorry but the truth is that you either have a wrong understanding of this specific subject (based on what I have read on this, feel free to prove me wrong) or you are using very imprecise language (claiming that something is inconsistent is substantively different from the claim that a system cannot prove its own consistency within itself - thats extremely imprecise language and thats being charitable towards you, because if any other person would make this big of a miscommunication, my assumption wouldn't be miscommunication, but my assumption would be that the other person lacks the understanding and thats why hes/she made a demonstrably false statement). Now, its obviously possible that I lack some contextual understanding on this, so feel free to prove my interpretation of the above texts wrong or feel free to point out how it ought to be interpreted and why.

Sure, but all of these limitations are directly applicable to your epistemology as well. Framing it this way is misleading, because my understanding is that both incompleteness theorems are only talking about provability and not about making "is" statements - so the system won't necessarily become inconsistent, it is just that you cannot prove within the system that the system itself is consistent (you need to go outside of the system in order to prove or disprove that system's consistency) - which is different from claiming that the system is inconsistent (because one is a truth claim, the other is a jusification[in this case investigating the limits of proving and proofs] and the truth value of a truth claim can be true or false regardless if I have the ability to prove it or disprove it). So in other words, from system's inability to prove its own consistency doesn't necessarily follow, that that system is actually inconsistent ( at least thats my understanding but I can be wrong).

Leo, you were changing your claims on the go as you realized ,your claims doesn't make sense or hold up. Starting with "all formal systems are contradictory" to "all formal systems with enough complexity are contradictory" to eventually making a difference between a contradiciton and incompleteness. I see no statement or implication of incompleteness here, I only see the assertion of inconsistency.

I think conflating those terms is a pretty big deal and shows a lack of understanding. Anyone who has a surface level understanding of basic logic wouldn't ever conflate incomplete with inconsistency . How can he make a video on this and not understand the meaning of those terms is baffling to me.

Im surprised that Leo would make such a statement that a finite system necessarily have to contain a contradiciton, when he did make a video about Gödel's incompleteness theorem himself. It seems that he either forgot or wasn't familiar with the "incomplete" option or he is being very vague and unprecise with his words again.

Im not sure what specific challenge you talk about. A Paradox comes with a specific challenge, but not all self referential systems entail a paradox , thats just a subset of the self referential systems.

I am familiar with the liar's paradox. Whats the argument you are trying to make?

A=2B 2A=4B A+2B =2A Show whats the "higher inconsistency that comes up here

"Look guys I have an idea how to revolutionize math, but I don't have an elementary understanding of math." How you manage to not cringe at your own self is impressive. You don't understand that 2^ 0=1 is not an axiom, its an entailment that comes from dividing a number by its own self equals to 1.

You don't have any understanding what powering a number to 0 even means. I already give you a breakdown and nemra gave you a breakdown also. I don't know why you are ignoring both of our answers Its extremely frustrating to engage with you , because you are willfully ignorant and you ignore answers and you trying to critique a subject that you don't even understand the basic concepts about.

IF you have 100 apples and divide it with 100 apples how much you get?

Dude you have to learn basic logic, of course you can create finite systems that are consistent.

(x^n)/(x^n)=x^0=1. This is an easy way to 'prove' it. If you divide a number with its own self you get one - is that different in this physical reality ?

wait, do you even know the justification for x^0=1? But again even if I grant you this, this doesnt really mean much to your original statement - the only conclusion follows from is that there are parts of math that cant be mapped onto physical reality, but it doesn't mean that current math is limited for modelling physical reality. As I already told you math is more than just about describing physical reality.

You are still confused about this. Some parts of math can be used to describe physical reality but there is much more to math than just describing physical reality. What do you specifically mean without being vague - that math isn't in alignment with phyiscs? So far when pushed on this you couldn't deliver anything tangible or of substance.

Whats next? Will you guys demand math to give prescription drugs for you?

You want to transform math to become physics at this point. I don't know why you would want to do that.