I was disappointed to find that adding together two equal infinite sets that are oppositely signed; for instance the real between -1 and -2 with the real numbers between 1 and 2 cannot be proven to equal zero. Common sense indicates it does.
ChatGPT said that the math is improvable presently. Someone should fix that since it seems like common sense.
ChatGPT -“Ah, thank you for the clarification! You’re asking about the intervals and , both subsets of the real numbers . These intervals are equal in cardinality (both are uncountably infinite). However, combining infinite sets algebraically, as you propose, doesn’t follow the same rules as arithmetic with finite numbers. Let’s explore the idea in detail.
ChatGPT -“Ah, thank you for the clarification! You’re asking about the intervals and , both subsets of the real numbers . These intervals are equal in cardinality (both are uncountably infinite). However, combining infinite sets algebraically, as you propose, doesn’t follow the same rules as arithmetic with finite numbers. Let’s explore the idea in detail.
Cardinality of the Sets
Both intervals and are uncountably infinite and have the same cardinality as the real numbers , denoted by (the cardinality of the continuum).
A bijection exists between these intervals (e.g., maps to bijectively), confirming their equivalence in “size.”
Algebra of Cardinalities
In the algebra of infinite cardinal numbers:
\mathfrak{c} – \mathfrak{c} = \mathfrak{c},
Measure Theory Interpretation
If you’re thinking about summing or subtracting these intervals with respect to their measures:
The measure (length) of is .
The measure (length) of is also .
In this case:
\text{Measure of } (1, 2) – \text{Measure of } (-2, -1) = 1 – 1 = 0.
But this applies only to the measure (length), not to the sets’ elements or their infinite nature.
Infinite Sets and Summation
If you’re suggesting that the “sum” of elements in the intervals should cancel out, this operation isn’t well-defined because:
The elements of the sets are infinite in number and uncountable, so “adding” them isn’t straightforward.
Even if we attempt to define something like , this wouldn’t “cancel out” infinite quantities but rather produce a value based on specific rules.
Conclusion
Your intuition about symmetry between the intervals is valid when considering their measure (length). However, in the domain of cardinality or infinite sets, subtraction doesn’t yield zero due to the unique properties of infinities in set theory. Let me know if you’d like further clarification or a specific framework (e.g., measure theory, integration, or set algebra) to explore!”
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