Comments (5)
This seems right to me; @timbutton ?
from openlogic.
@rzach yes, this needs fixing, and I appreciate @pglutz's suggestion as to how.
I am frantic with admin tasks for the next two weeks, so won't be able to fix it for a little while. @rzach, feel free to fix it if you like; otherwise, I'll get to it in mid-February.
from openlogic.
Is the proposal to change the definition of ℚ to be equivalence classes of pairs [a,b] where a, b ∈ ℤ but b is positive? I wonder if an easier (ie, more local) fix would be to leave the definition of ℚ the same and just redefine ≤ on ℚ as
[a,b] ≤ [c,d] iff [c,d] - [a,b] ~ [e_ℤ,f_ℤ] for some e ∈ ℕ, 0 ≠ f ∈ ℕ
(that is, p ≤ q iff q-p is not negative).
I think nothing else has to be changed since the verification that ℚ is an ordered field is left as an exercise. I don't know if that exercise would be harder with this definition of ≤ (where only the verification of the order properties would change) or @pglutz 's (where all the field properties would change).
from openlogic.
@rzach I think you misunderstood the fix I suggested but your fix is probably better than what I had in mind anyway.
In case you're curious, the fix I had in mind was to leave the definition of Q the same but change the definition of the order to:
[a, b] \leq [c, d] iff for some [a', b'] ~ [a, b] and some [c', d'] ~ [c, d] we have b' > 0, d' > 0 and a'd' \leq c'b'.
This does not require changing the definition of Q but it is more of a mouthful and to show it defines an order requires justifying why every ~ equivalence class [a, b]_~ contains an element [a', b'] such that b' > 0.
Your fix is nice because it seems a bit cleaner and less ad-hoc.
from openlogic.
from openlogic.
Related Issues (20)
- Multiple incompatible definitions for "total order" HOT 1
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- Definition 1.1 (Extensionality). If A and B are sets, then A = B iff every element of A is also an element of B, and vice versa. HOT 2
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- Typo in Lindenbaum's lemma proof HOT 1
- Suggested improvements for the proof of Lindenbaum's lemma HOT 3
- Minor error in example 12.25 HOT 2
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- Possible typo in example 14.14
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from openlogic.