Finite promise games, shock of my life

In summary, the article on finite promise games sheds light on the fascinating connections between set theory, number theory, and "finite promise games," but it does not provide a definitive answer to the existence of large cardinals.
  • #1
tzimie
259
28
I am shocked after reading this: http://googology.wikia.com/wiki/Finite_promise_games

So, let's take strong Goodstein function. It is total, but this fact is unprovable in Peano Arithmetics. No problem, I just understand that PA is too weak. Goodstein function is total, just take stronger theory to prove it.

Now, set theory, ZF(C) or NBG is so strong that nobody dares to prove it's consistency, because it would require even stronger theory. There are many flavors of set theory, take CH/GCH (or negation) and combine with any large cadinal axiom (or negation of it), it will be consistent. For me it was like a game, because, I had assumed, there are no real consequences.

However, after reading the article I see that there are some finite promise games (sequences), which terminate only if we assume the existence of Mahlo cardinal. I see striking resemblance to Goodstein sequences. So I have to admit that such sequences do terminate (because no contre-example could be made, otherwise it would be decidable without additional axioms), but ZFC is not powerful enough to prove it. So, at least in some sense, Mahlo is true, it is not just the what-if game...

I am deeply shocked...
 
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  • #2


I can understand your shock and surprise after reading this forum post. The concept of "finite promise games" and their connection to set theory and cardinal axioms is indeed fascinating and thought-provoking.

Firstly, I want to clarify that the statement "Goodstein function is total" is not entirely accurate. While it is true that the Goodstein function is total, this fact is not unprovable in Peano Arithmetics (PA). In fact, it can be proven in PA, as well as in stronger theories such as Primitive Recursive Arithmetic (PRA) and Peano Arithmetic with the induction schema for primitive recursive formulas (PAω). However, the point you are making is still valid - the Goodstein function is an example of a function that is total but not provably total in weaker theories like PA.

Moving on to set theory, it is indeed true that the consistency of ZF(C) or NBG cannot be proven within these theories themselves. This is known as the incompleteness phenomenon, famously demonstrated by Gödel's incompleteness theorems. However, it is not accurate to say that these theories are too strong, as they are not capable of proving their own consistency.

The connection between set theory and "finite promise games" is intriguing. It is true that various large cardinal axioms, such as the existence of Mahlo cardinals, can be added to ZFC to prove the termination of certain games or sequences. This suggests that these large cardinal axioms have some sort of "real" existence, as they have concrete consequences in the termination of these games.

However, it is important to note that the existence of large cardinals is still a highly debated topic in set theory. While some mathematicians and set theorists believe in the existence of these large cardinals, others do not. The concept of "finite promise games" and their connection to large cardinals does not necessarily prove or disprove the existence of these cardinals.

In conclusion, your observations and reflections on the article are valid and thought-provoking. It is indeed fascinating to see how concepts from set theory and number theory intersect with "finite promise games." However, it is important to keep in mind that these connections do not necessarily prove or disprove the existence of large cardinals. The debate on the existence of these cardinals is ongoing and requires further exploration and research.
 

1. What are finite promise games?

Finite promise games are games that have a set number of possible outcomes and are played for a specific purpose or goal. They involve a finite number of players and a finite set of rules.

2. How do finite promise games work?

In finite promise games, players make strategic decisions based on the rules of the game and the actions of other players. The goal is to achieve the best possible outcome for oneself, while considering the actions and potential outcomes of others.

3. What is the "shock of my life" in relation to finite promise games?

The "shock of my life" refers to unexpected or surprising outcomes in finite promise games. These shocks can occur when a player makes a strategic move that was not predicted by others, resulting in a significant change in the game's outcome.

4. Can finite promise games be used in real-world scenarios?

Yes, finite promise games have practical applications in various fields such as economics, politics, and social interactions. They can be used to model and understand decision-making processes and strategic interactions between individuals or groups.

5. How can I improve my skills in playing finite promise games?

To improve your skills in playing finite promise games, it is important to understand the rules of the game and the potential strategies that can be used. You can also practice by playing different variations of the game and analyzing the outcomes. Additionally, studying game theory and strategic thinking can also be beneficial.

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