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May 16

Is there any mathematical formulation exist for Grandfather paradox?

Is there any mathematical formulation for Grandfather paradox? In the article, I didn't find any formulation. If there is mathematical formulation for EPR paradox why not this? Rizosome (talk) 15:59, 16 May 2021 (UTC)[reply]

It's not a mathematical concept. The article very well describes what it is, and asking why there isn't a mathematical formulation to it is like asking why there isn't a mathematical formulation to Romeo and Juliet. --OuroborosCobra (talk) 18:31, 16 May 2021 (UTC)[reply]

Don't be so sure on that. 3Blue1Brown has done mathematical formulations of Romeo and Juliet for his videos. I believe he did it for one of his differential equation videos. --Jayron 32 11:56, 17 May 2021 (UTC)[reply]

In the context of special relativity, there exists the example of the two-way tachyonic antitelephone. So, Alice is moving away from Bob at speed of v c where c is the speed of light. She sends a message to Bob using tachyons that travel at a speed of a c with a >1, and Bob immediately sends back the message to Alice using tachyons. Then, as shown in the article, if v > 2 a/(1+a^2) which is always possible for v < 1, Alice will receive her message back from Bob before she sent her original message to Bob. Count Iblis (talk) 09:04, 17 May 2021 (UTC)[reply]

I don't know why the paradox is made unnecessarily complicated by involving three generations and the uncertainty surrounding biological fatherhood. In a much simpler version of the paradox, someone kills their mother before they were borm. Or, told in a story:

On a particularly bad day, Cain is seething with anger and kills a man he encounters, the famous mathematician and time traveller Niels Abel, who came to see if the Biblical account of Cain killing his brother was factual. Cain discovers the traveller's time machine, presses a button and is transported back in time to even before Eve knew Adam in the Biblical sense. Still angry, Cain kills his mother-to-be. Adam, her widower, is left alone with Cain. A remorseful Cain travels to the 19th century to warn Abel not to travel to the past and be killed, but finds the Earth unpopulated.

This can be modelled as a paradox in predicate logic enriched with an ordered time domain. We introduce three predicates, B, K and L, with the following intuitive interpretations:

B(P,Q,T) means: person P gives birth to person Q at time T;

K(P,Q,T) means: person P kills person Q at time T;

L(P,T) means: person P is alive at time T;

For each, we have an axiom:

(ax_B) B(P,Q,T) → L(P,T)

(ax_K) K(P,Q,T) → ¬L(Q,T)

(ax_L) ¬L(P,T₁) ∧ T₁ < T₂ → ¬L(P,T₂)

Intuitively, the first axiom states that only a live mother can give birth, the second that a person who is killed is dead, and the third denies the option of resurrection from the dead. To model the paradox, we need to introduce person constants Eve and Cain, two time constants T1 and T2, and three premises:

(1) K(Cain,Eve,T1);

(2) B(Eve,Cain,T2);

(3) T1 < T2.

Now we reason

(4) K(Cain,Eve,T1) → ¬L(Eve,T1) (by instantiation of (ax_K))

(5) ¬L(Eve,T1) (by modus ponens from (4) and (1))

(6) B(Eve,Cain,T2) → L(Eve,T2) (by instantiation of (ax_B))

(7) L(Eve,T2) (by modus ponens from (6) and (2))

(8) ¬L(Eve,T1) ∧ T1 < T2 → ¬L(Eve,T2) (by instantiation of (ax_L))

(9) ¬L(Eve,T1) ∧ T1 < T2 (by ∧-introduction from (5) and (3))

(10) ¬L(Eve,T2) (by modus ponens from (9) and (8))

We have now derived both L(Eve,T2) and ¬L(Eve,T2); the conclusions (7) and (10) contradict each other. --Lambiam 11:00, 17 May 2021 (UTC)[reply]

But note that our article says:

"Despite its title, the grandfather paradox does not exclusively regard the contradiction of killing one's own grandfather to prevent one's birth. Rather, the paradox regards any action that alters the past, since there is a contradiction whenever the past becomes different from the way it was".

A fictitious example is in A Sound of Thunder (1952) in which some time travellers return from the age of dinosaurs to a changed reality, having accidentally trodden on a butterfly. Alansplodge (talk) 11:12, 17 May 2021 (UTC)[reply]

The paradox also requires the notion that the future cannot be altered. The Bradbury story is an early example of exploring the idea of alternate timelines branching off from a shared past, so there is no paradox. Also without paradox, in a universe with a single time strand, it might have (fictionally) been the case that Deutscher's initial election loss was a consequence of the butterfly being crushed, the two events being connected by a causal chain spanning some 66,000 millenia. In that case, indeed, the past has never been altered; it has always been and remained as it was, and likewise for the future. The logic model above assumes an ordered time domain, excluding alternate timelines. (In computation tree logic there is no inconstency.) But if the model does not allow for alternates, it does not make sense (IMO) to say that the paradox arises from the past "becoming" different from the way it "was". The logic does not model "changes"; the notion of the past being changed arises from our way of forcing something inconsistent into a seemingly coherent narrative. --Lambiam 20:36, 17 May 2021 (UTC)[reply]