[FOM] Coins and Infinite Ordinals

Dmytro Taranovsky dmytro at mit.edu
Sun Jul 20 13:51:32 EDT 2014

General knowledge of infinite ordinals and ordinal notation systems has 
been hampered by their complexity and difficulty of visualization.  Here 
I present an easily visualizable description using something that most 
people know in connection with finite ordinals -- coins.

The general framework is that one's wealth (which represents an ordinal) 
is a finite set of coins.  A coin can be exchanged with (equivalently, 
is more valuable than) any finite number (including zero) of less 
valuable coins.  Thus, a coin corresponds to an ordinal that is a power 
of omega, and comparison of wealth reduces to comparison of coins.  
(Coins in other denominations are not discussed here.)  Well-foundness 
means that if you keep spending and exchanging coins with less valuable 
ones without replenishing, then you will eventually run out of money.

-- Up to epsilon_0 --

* A coin depicts some (possibly zero) wealth on its face.
* A coin that depicts more wealth is more valuable, and two coins that 
depict the same amount of wealth are equivalent.

Thus, a plain coin corresponds to 1, a coin depicting 2 plain coins is 
worth omega^2, and in general, a coin depicting wealth x is worth omega^x.

-- Bachmann-Howard Ordinal --

* A coin can be made of silver or gold and depicts some (possibly zero) 
* Gold coins are more valuable than silver.
* For coins made of the same material, coins that depict more wealth are 
more valuable (and two coins made of the same material and depicting the 
same amount of wealth are equivalent).
* A silver coin may not depict a silver coin of greater value. Note: 
Here 'depict' is transitive; a coin may depict another coin indirectly 
through intermediate coin(s).

Bachmann-Howard ordinal Omega (or if we insert natural gaps in the 
notation system, the least nonrecursive ordinal) corresponds to a plain 
gold coin.  A gold coin depicting wealth x is worth omega^(Omega+x).  A 
silver coin depicting wealth x is worth omega^(collapse of x) for an 
appropriate ordinal collapsing function.  Also, because we have two 
materials, coins can now depict more valuable coins, which necessitates 
some restriction against infinite regress of coin values.

-- Pi^1_1-CA_0 --

By iterating the construction for Bachmann-Howard ordinal, we get a 
notation system for Pi^1_1-CA_0 (same strength as KP + {there are n 
admissible ordinals : n<omega}).

* Coins are classified by the material (a natural number) and the amount 
of wealth depicted.
* If m<n, then a coin made from material m is worth less than a coin 
made from material n.
* For coins made from the same material, coins that depict more wealth 
are worth more.
* A coin made from material n may not directly (as opposed to using 
intermediate coin(s)) depict a coin of material above n+1. 
(Equivalently, the total wealth depicted must be less than a plain coin 
made from material n+2.)
* A coin made from material n may not depict a more valuable coin made 
from material n, except when it (the more valuable coin) is inside a 
coin made from a material less than n.

-- Beyond --

Stronger ordinal notation systems can be made using just silver and gold 
coins (gold coins more valuable than silver; coins from the same 
material classified and ordered by the amount of wealth depicted), but 
with more complicated rules about when a coin can depict a coin of 
greater value.

-- Visualization --

While the description may be clear for FOM readers, nontechnical people 
would benefit if there was a software system that allowed them to mint 
and validate ordinal coins, see the coins on the screen, exchange coins, 
and do other operations that will make the coins appear simple and 
concrete.  For example, one useful operation is combining wealth by 
pooling two sets of coins together; this correspond to symmetric ordinal 
sum, which, unlike ordinary ordinal sum, is commutative.

Dmytro Taranovsky

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