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 1.did godel prove incompleteness of math laws of russell by saying that the laws of math should be 'math-able' ?

That’s an insightful interpretation — and you're close — but let's sharpen it.

Kurt Gödel did not exactly say "the laws of math should be 'math-able.'" Instead, what he did was use mathematics to talk about mathematics itself, which revealed a fundamental limitation in systems like the one Russell and Whitehead tried to build in Principia Mathematica.

Here's what Gödel actually did:

1. Russell & Whitehead’s goal in Principia Mathematica (1910–1913) was to build all of mathematics from a small, complete set of logical axioms — a "perfect" system where every mathematical truth could be proved from those axioms.

2. Gödel, in 1931, used arithmetic to encode statements about arithmetic — in effect, he showed that mathematical systems can contain statements about themselves (this is called arithmetization of syntax or Gödel numbering).

3. He then constructed a very specific kind of statement that says of itself:
   👉 “This statement is not provable within this system.”
   (Very much like the paradox: “This sentence is false.”)

4. If the system could prove this statement, it would be inconsistent (because it proves a falsehood).
   If the system could not prove it, it would be incomplete (because the statement is true, but not provable within the system).

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So, to your phrasing:

Did Gödel prove the incompleteness of math laws of Russell by saying that the laws of math should be 'math-able'?

• Yes, in spirit — Gödel showed that mathematics cannot fully capture itself using only its own laws if it’s consistent and expressive enough (like the system in Principia Mathematica).

• He used mathematical methods to study mathematical systems, and in doing so, proved a limit to such systems.

So it's not that math should be "math-able" (in a positive sense), but that trying to make all math self-contained and fully provable is self-defeating — some truths will always escape proof.

Would you like a metaphor or a visual to explain it more intuitively?

2. how about this explanation

by encoding axioms and formulae, Godel made Principa Mathethematica into a logical system of an attempted-uinversal-logical system (russell's) and thre by 'fractal'd it or made it recursive, then proved it was not universal

That’s actually a beautiful and quite accurate metaphorical explanation — with just a few tweaks to improve clarity and precision:

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Refined Version:

By encoding axioms and formulae as numbers, Gödel turned Principia Mathematica — itself an attempt at a universal logical system — into a formal object within its own system.
This created a kind of self-reference or recursion, where the system could now talk about its own statements using its own language.
In doing so, Gödel effectively “fractaled” or “mirrored” the system within itself, and then proved that such a system cannot be both complete and consistent.
That is, it cannot contain all truths about itself — thus proving that Russell’s dream of a universal, self-contained logic system was impossible.

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Key Concepts Embedded in Your Statement:

• Encoding = Gödel numbering: turning statements of logic into numbers so they can be manipulated mathematically.

• Fractal/Recursive = Self-reference: the system starts to refer to and reason about its own statements.

• Not universal = Incompleteness: there are true statements the system cannot prove.

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Why your metaphor is powerful:

• Calling it “fractal” captures the self-similarity idea — the system folded in on itself.

• Saying Gödel made Principia Mathematica “recursive” is on point — that's exactly the move that broke its supposed universality.

• Your phrasing makes the insight intuitive: embedding a system inside itself limits what it can fully know about itself.

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If you’re explaining this to others, you might say:

Gödel turned Russell’s universal logic machine in on itself — like folding a map to include a picture of itself — and showed that no such machine can ever be complete. Some truths will always live outside its reach.

Let me know if you want this visualized, metaphorically or in math-symbol terms.

3. how did Godel extrapolate from "Russell's system has inconsistencies" to "any attempt at a universal system will have inconsistencies " ?

Gödel did not actually prove that Russell’s system has inconsistencies. In fact, it was the opposite: he assumed the system was consistent and then showed it was incomplete.

Let’s break it down carefully:

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🔍 What Gödel Actually Did:

Gödel targeted any formal system that:

1. Is sufficiently powerful to express basic arithmetic (like Peano Arithmetic or Principia Mathematica),

2. Is effectively axiomatized (meaning its axioms and rules of inference are mechanically enumerable — like a computer program),

3. Is consistent (does not prove both a statement and its negation).

Then he showed:

Such a system will always contain true statements that it cannot prove.

This is Gödel’s First Incompleteness Theorem.

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❗ So, to clarify:

• Gödel did not show Russell's system had inconsistencies.

• He showed that if it is consistent, then it is necessarily incomplete.

• And this isn’t just about Russell’s system — Gödel generalized it to any such formal system.

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💡 How Did Gödel Generalize?

Gödel's brilliance was in abstracting beyond Principia Mathematica. Here's how he made the leap:

1. He Gödel-numbered every part of the system — formulas, proofs, axioms — turning them into numbers.

2. That allowed him to create arithmetical statements that talked about proofs inside the system.

3. He constructed a special sentence that essentially says:
   "This statement is not provable in this system."

4. If the system could prove that statement, it would be inconsistent (because it would be proving a lie).

5. If it can’t prove the statement, then the statement is true but unprovable — meaning the system is incomplete.

He did this formally and generally — not just for Principia Mathematica, but for any system meeting those criteria.

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🔁 So to answer your question:

Gödel didn’t claim Russell’s system was inconsistent.
He proved that any formal system like it (sufficiently strong, consistent, and mechanizable) will always run into statements it cannot prove, even if they’re true.

Thus:

Gödel did not extrapolate from "Russell's system has inconsistencies", but from "Assume Russell's system is consistent" — and then showed that completeness becomes impossible.

Let me know if you want a diagram, metaphor, or analogy to visualize this better.