[FOM] 756: Large Cardinals and Emulations/36
Harvey Friedman
hmflogic at gmail.com
Fri Mar 24 08:03:48 EDT 2017
We continue with the organized presentation of Emulation Theory.
EMULATION THEORY
Below will organize, as FOM postings, the contents of a ms. to be
placed on my website, and to appear in some form in a planned volume
in honor of Putnam. This ms. will contain proofs other than reversals.
Reversals will appear in a much expanded ms. later as a high priority,
and will form the book CONCRETE MATHEMATICAL INCOMPLETENESS, when
combined with the BOOLEAN RELATION THEORY ms. currently on my website.
1. INTRODUCTION.
http://www.cs.nyu.edu/pipermail/fom/2017-February/020299.html
2. DROP EQUIVALENCE IN LINEAR ORDERINGS.
http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.html
3. MAXIMAL EMULATION IN Q[0,1]^k
http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.html
3,1, DROP EQUiVALENCE.
http://www.cs.nyu.edu/pipermail/fom/2017-February/020301.html
3.2. USABILITY.
Large Cardinals and Emulations/34
3.3. EMBED USABILITY.
Large Cardinals and Emulations/35
continued here
3.4. TRANSLATION USABILITY.
4. STEP MAXIMAL EMULATION IN Q^k.
5. SUBMAXIMAL EMULATION IN Q^k.
6. GREEDY EMULATION IN Q^k.
7. UP GREEDY EMULATION IN Q^k.
8. FINITE EMULATION.
8.1. WEAKLY MAXIMAL EMULATION IN Q^[0,1]^k
8.2. WEAKLY STEP MAXIMAL EMULATION IN Q^k.
8.3. WEAKLY SUBMAXIMAL EMULATION IN Q^k.
8.4. WEAKLY GREEDY EMULATION IN Q^k
8.5. WEAKLY UP GREEDY EMULATION IN Q^k.
NOTE: HUGE cardinals appear in sections 7 and 8.5. In section 8.1, we
will see this:
For finite subsets of Q[0,1]^k, some weakly maximal finite r-emulation
is drop equivalent at (1,1/2,...,1/k),(1/2,...,1/k,1/k).
with estimates taking it from explicitly Pi02 to explicitly Pi01. And
similarly for 8.2-8.4. In section 7, we contrast
For finite subsets of Q^k, some (up) greedy emulation contains its
upper shift. Con(SRP).
For finite subsets of Q^k, the lower sections of some up greedy
emulation include those of its upper shift. Con(HUGE).
In 8.5, find explicitly Pi01 statements equivalent to Con(HUGE).
3.3. EMBED USABILITY
continued from Large Cardinals and Emulations/35
Now that we have completely determined the ME embed usable finite
functions, further major steps in the development of Emulation Theory
should include the determination of the ME embed usable functions
f::Q[0,1] into Q[0,1]. Note that such a determination would include
MED/1 as a very special case, as MED/1 assets that the function
h_k::Q[0,1] into Q[0,1] given by
h_k(p) = p if p in Q[0,1/k); f(1/i) = 1/(i+1) if i = 1,...,k-1;
undefined otherwise
is ME embed usable.
A far more realistic and very natural project is to determine the
embed usability of the order theoretic f::Q[0,1] into Q[0,1], as
defined in the Introduction.
THEOREM ?. f::Q[0,1] into Q[0,1] is order theoretic if and only if f
is the finite disjoint union of constant and identity functions on
intervals in Q[0,1]. The inverse of one-one order theoretic f::Q[0,1]
into A[0,1] is order theoretic.
Recall from the Introduction that intervals in Q[0,1] are of the form
[(p,q)], p,q in Q[0,1], and degenerate intervals are those which are
empty or singletons. Note that the h_k above are order theoretic.
We have not been able to determine the embed usability of order
theoretic f::Q[0,1] into Q[0,1]. However, we have some significant
partial results. Note that the h_k
DEFINITION 3.3.5. f::Q[0,1] into Q[0,1] is continuous if and only if
for all p in dom(f) and epsilon > 0, there exists delta > 0 such that
for all q in dom(f) with |p-q| < delta, we have |f(p)-f(q)| < epsilon.
f::Q[0,1] into Q[0,1] is bicontinuous if and only if f is one-one and
f,f^-1 are continuous.
We extend MEE/2 to bicontinuous order theoretic functions. (There is a
typo in MEE/2. "ME usable" should be "ME embed usable").
MAXIMAL EMULATION EMBEDDING/3. MEE/3. A bicontinuous order theoretic
f::Q[0,1] into Q[0,1] is ME embed usable if and only if it is strictly
increasing, and
i. There is no 0 < p,q < 1 such that f maps 0 to p and q to 1.
ii. There is no 0 < p,q < 1 such that f maps p to 0 and 1 to q.
We will prove MEE/3 in ZC. It is likely that MEE/3 is not provable in
WZC (weak ZC which is ZC with bounded separation). We also extend
MEE/3 as follows..
MAXIMAL EMULATION EMBEDDING/4. MEE/4. A continuous dominating order
theoretic f::Q[0,1] into Q[0,1] is ME embed usable if and only if it
is strictly increasing, and
i. There is no 0 < p,q < 1 such that f maps 0 to p and q to 1.
ii. There is no 0 < p,q < 1 such that f maps p to 0 and 1 to q.
Recall dominating means p in dom(f) implies f(p) >= p.
We will show that MEE/4 is provably equivalent to Con(SRP) over WKL_0.
Here are three stronger statements whose status is unknown to us.
MAXIMAL EMULATION EMBEDDING/5. MEE/5. A continuous order theoretic
f::Q[0,1] into Q[0,1] is ME embed usable if and only if it is strictly
increasing, and
i. There is no 0 < p,q < 1 such that f maps 0 to p and q to 1.
ii. There is no 0 < p,q < 1 such that f maps p to 0 and 1 to q.
MAXIMAL EMULATION EMBEDDING/5. MEE/6. A dominating order theoretic
f::Q[0,1] into Q[0,1] is ME embed usable if and only if it is strictly
increasing, and
i. There is no 0 < p,q < 1 such that f maps 0 to p and q to 1.
ii. There is no 0 < p,q < 1 such that f maps p to 0 and 1 to q.
MAXIMAL EMULATION EMBEDDING/5. MEE/7. An order theoretic f::Q[0,1]
into Q[0,1] is ME embed usable if and only if it is strictly
increasing, and
i. There is no 0 < p,q < 1 such that f maps 0 to p and q to 1.
ii. There is no 0 < p,q < 1 such that f maps p to 0 and 1 to q.
We have already established the forward directions of MEE/3-7 in
RCA_0, as strictly increasing, and i,ii are necessary for any ME embed
usable f::Q[0,1] into Q[0,1].
We do not know anything more about the status of MEE/5-7 other than that
they immediately imply MEE/4, and therefore imply Con(SRP) over
WKL_0.
DEFINITION 3.3.6. Let f_1,...,f_n,g_1,...,g_n::Q[0,1] into Q[0,1].
(f_1,...,f_n), (g_1,...,g_n) are Q[0,1] isomorphic if and only if
there is a strictly increasing bijection h:Q[0,1] into Q[0,1] such
that for all p in dom(f_i), f_i(p) = q iff g_i(hp_ = hq.
(f_1,...,f_n), (g_1,...,g_n) are reverse isomorphic if and only if
there is a strictly decreasing bijection h:Q[0,1] into Q[0,1] such
that for all p in dom(f_i), f_i(p) = q iff g_i(hp) = hq. I.e., in both
cases, h is an isomorphism from (Q[0,1],f_1,...,f_n) onto
(Q[0,1],g_1,...,g_n), and in the first case, an isomorphism from
(Q[0,1],<,f_1,...,f_n) onto (Q[0,1],g_1,...,g_n,<).
The following should appear earlier in the ms. In fact, a version for
usability should be in section 3.2, with the following derived as a
consequence.
THEOREM 3.3.7. Let f_1,...,f_n,g_1,...,g_n::Q[0,1] into Q[0,1], where
(f_1,...,f_n), (g_1,...,g_n) are isomorphic or reverse isomorphic.
Then (f_1,...,f_n) is ME embed usable if and only if (g_1,...,g_n) is
ME embed usable.
Proof: Let h be an isomorphism from (Q[0,1],<,f_1,...,f_n) onto
(Q[0,1],<,g_1,...,g_n), and suppose (f_1,...,f_n) is ME embed usable.
Let E containedin Q[0,1]^k and r >= 1. Let S be a maximal r-emulation
of E which is (f_1,...,f_n)-invariant. Since this is a first order
property of (Q[0,1],<,S,f_1,...,f_n), it is preserved under the action
of h, and so it holds of (Q[0,1],<,hS,g_1,...,g_n). So hS is a maximal
r-emulation of E which is (g_1,...,g_n)-invariant.
Now let h be a reversing isomorphism from (Q[0,1],<,f_1,...,f_n) onto
(Q[0,1],<,g_1,...,g_n), and suppose (f_1,...,f_n) is ME embed usable.
Let E containedin Q[0,1]^k and r >= 1. Let S be a maximal r-emulation
of E which is (f_1,...,f_n)-invariant. Since this is a first order
property of (Q[0,1],<,S,f_1,...,f_n), it is preserved under the action
of h, and so it holds of (Q[0,1],>,hS,g_1,...,g_n). So hS is a maximal
r-emulation of E which is (g_1,...,g_n)-invariant, in the sense of
maximal r-emulation that uses > in place of <. But the notion of
maximal r-emulation using < as usual, is equivalent to the notion of
maximal r-emulation using > in place of <. QED
NOTE: Already in the Introduction, probably should prove this last
claim that < can be replaced by >.
LEMMA 3.3.8. (RCA_0) Let f::Q[0,1] into Q[0,1] be order theoretic,
strictly increasing, and bicontinuous. Then f can be extended to a
strictly increasing bicontinuous function which is Q[0,1] isomorphic
or reverse isomorphic to a function of the form G U. h, where
i. G is the identity on [0,1/2n] U [1/(2n-1),1/(2n-2)] U ... U
[1/3,1/2] and h is finite, h alters 1, n >= 1. (In case n = 1, there
is only the lead term).
ii. G is the identity on [0,1/(2n+1)] U [1/(2n),1/(2n-1)] U ... U
[1/2,1] and h is finite, n >= 1.
iii. G is the identity on [1/(2n+1),1/(2n)] U [1/(2n-1),1/(2n-2)] U
... U [1/3,1/2] and h is finite, h alters 0 and 1.
iv. G is the identity on Q[0,1].
v. G,h are finite.
Proof: Let f be as given. By Theorem ?, f is the disjoint finite union
of strictly increasing constant and identity functions on intervals in
Q[0,1]. Hence write f = G U. h, where G is the identity function on a
disjoint union of nondegenerate intervals and h is a finite function.
If h is empty then extend f to the identity on Q[0,1]. If G is empty
then f is finite. Assume f,G are nonempty.
Since f is bicontinuous, h is the identity on all endpoints of these
nondegenerate intervals that lie in fld(h). So we remove these (p,p)
from h and add them to G, and also extend G by the identity on the
endpoints of its nondegenerate intervals. We now have f' = G' U. h',
where f' extends f, f' is strictly increasing and bicontinuous, G' is
the identity on a finite disjoint union of closed intervals, and h' is
finite.
There are four cases of whether h does/does not alter 0, and h
does/does not alter 1. If h does not alter 0 then further extend by
the identity on a nondegenerate closed interval containing 0, disjoint
from the present field. If h does not alter 1 then we can further
extend by the identity on a nondegenrate closed interval containing 1
disjoint from the present field.
For three of these four cases, we clearly arrive at a function
isomorphic to a function in categories i,ii,iii. For the fourth case,
we arrive at a function reverse isomorphic to a function in category
i. QED
LEMMA 3.3.9. (ZC) Every f of category i in Lemma 3.3.8 is ME embed usable.
Proof: Let f be in category i, f = G U. h, f strictly increasing, G is
the identity on [0,1/2n] U [1/(2n-1),1/(2n-2)] U ... U [1/3,1/2] and h
is finite, h alters 1, n >= 1. (In case n = 1, there is only the lead
term). Let E containedin Q[0,1]^k and r >= 1. We construct a maximal
r-emulation of E that is f-embedded. Let t = |fld(h)|.
Let D = {1,...,n} x lambda x Q[0,1) and <D be the lexicographic
ordering on D using the usual < on the three factors. I.e., first by
the integer, second by the ordinal, and third by the rational. But
(D,<) is not a well ordering, and we need an associated well ordering
of D. Let Q[0,1)' be any ordering of Q[0,1) of type omega where the
least element is 0. Let <# be the well ordering of D = {1,...,n} x
lambda x Q[0,1)' first by the ordinal, second by the integer, and
third by the rational (using Q[0,1]'). Finally, let <#^k be the well
ordering of D^k first by the <# of the k coordinates, and then
lexicographically by the <# of the k coordinates.
Now let S* containedin D^k be the maximal r-emulation of E which is
<#^k greedy. Note that S* is definable over (V(lambda),epsilon). Let
kappa_1 < ... kappa_tn be first order indiscernibles over
V(lambda,epsilon). Now ZC does not prove the existence of even the
ordinal omega + omega, and so some argument is needed that this proof
can be carried out in ZC. Suffice it to say that we are using the
Erdos Rado theorem which involves approximately tn iterations of the
power set operation over V(omega).
We now want to lift f::Q[0,1] into Q[0,1] to f*::D into D. We lift
dom(G) to dom(G)*, G to G*, fld(h) to fld(h)*, h to h*, and finally f
to f*. G is the disjoint union of n nondegenerate closed intervals in
Q[0,1] with 2n successive endpoints starting with 0. Assign elements
of D to these successive endpoints as follows. The left endpoints get
assigned (1,0,0), (2,0,0), ..., (n,0,0). The right endpoints get
assigned (1,1,0), (2,1,0), ..., (n,1,0). dom(G)* is the union of the
corresponding closed intervals in (D,<D). G* is the identity on
dom(G)*. Now assign the initial segment of (i,kappa_0,1+(i-1)t),
(i,kappa_1,0), ..., (i,kappa_it) to the elements of fld(h) between the
i-th and (i+1)-th interval of dom(G), 1 <= i < n, in increasing order,
and the tail of (n,kappa_1+(n-1)t,0), ..., (n,kappa_tn,0) to the
elements of fld(h) above the last interval of dom(G). fld(h)* is the
set of all of these assignments. Now lift h to h*::fld(h)* into
fld(h)* canonically. Clearly fld(h*) = fld(h)*. Finally, define f* =
G* U. h*.
We claim that our maximal r-emulation of E, S containedin D^k, is
f*-embedded. To see this, consider any k-tuple from dom(G*) and the
elements of dom(h*). When we apply f*, we get the same coordinates
from dom(G*) and we get various coordinates from rng(h*). For the
latter, we preserve the integer terms, preserve the rational terms 0,
but generally move the ordinal term, which was an indiscernible and
remains an indiscernible. The relative order of these ordinal terms is
preserved since h is strictly increasing. Hence by the
indiscernibility, membership in S* remains the same.
Now look at the system (D,<D,S*,f*), We want to go back down to the
rationals and form the system (Q[0,1],<,S,f), where we have yet to
define S. Note that <D is a dense linear ordering with left endpoint
(1,0,0) but with no right endpoint. We let D* be the elements of D
whose ordinal is < kappa_nt, together with those whose ordinal is
kappa_nt and whose integer is <n, together with (n,kappa_nt ,0), and
<D* be <D restricted to D*. Note that <D* is a dense linear ordering
with endpoints (1,0,0), (n,kappa_nt,0), and is also the initial
segment of <# up through (n,kappa_nt,0). Note that f*::D* into D*.
Also by the greedy construction of S*, we see that S* intersect D*^k
containedin D*^k is a maximal r-emulation of E.
We now form (D*,<D*,S* intersect D*^k,f*). Note that fld(f*) is
countable, and "<D is dense" and "S* intersect D*^k" is a maximal
r-emulation of E that is f*-embedded" are AE assertions. Therefore by
an inductive construction of length omega, we can form the subsystem
(D**,<D**,S* intersect D**^k,f*), f**::D** into D**, where D** is
countable, <D** is countable dense with the same endpoints, and S*
intersect D**^k containedin D**^k remains a maximal r-emlation of E
that is f*-embedded. We now use any increasing bijection from <D**
onto Q[0,1] to form the isomorphic copy (Q[0,1],<,S,f). S containedin
Q[0,1]^k will be a maximal r-emulation of E that is f-embedded since
AE sentences are preserved. QED
LEMMA 3.3.10. (ZC) Every f of category ii in Lemma 3.3.8 is ME embed usable.
Proof: The same as for Lemma 3.3.9 with some minor modifications. D,
<D, Q[0,1)', <#, <#^k, S*, kappa_1,...,kappa_tn are defined in exactly
the same way except that we use {1,...,n+1} instead of {1,...,n}. We
lift f* in the same way where there is the extra final closed interval
in dom(f*) at the end, which is [(n+1,0,0),(n+1,1,0)].
We let D* be the elements of D whose integer is <= n, whose ordinal is
<= kappa_nt, together with the closed interval [(n+1,0,0),(n+1,1,0)]),
and <D* be <D restricted to D*. Note that <D* is a dense linear
ordering with endpoints (1,0,0), (n+1,1,0), and is also the initial
segment of <# up through (n+1,1,0). We lift f to f*::D* into D* as
before. WE form (D*,<D*,S* intersect D*^k,f*) and the countable
subsystem (D**,<D**,S* intersect D**^k,f*), f*:: D** into D**, as
before. We then use any increasing bijection from <D** onto Q[0,1] as
before to form the isomorphic copy (Q[0,1],<,S,f), S containedin
Q[0,1]^k being a maximal r-emulation of E that is f-embedded. QED
For handling category iii in Lemma 3.3.8, it is convenient to shift
from Q[0,1] to Q[-1,1].
DEFINITION 3.3.7. f::Q[-1,1] into Q[-1,1] is symmetric if and only if
for all p in dom(f), f(-p) = -f(p).
LEMMA 3.3.11. (RCA_0) Every f of category iii in Lemma 3.3.8 with
(f(0) > 0 and f(1) < 1) or (f^-1(0) > 0 and f^-1(1) < 1) extends to an
f' in category iii in Lemma 3.3.8 which is isomorphic to a symmetric
g::Q[-1,1] into Q[-1,1] with 1/2 not in fld(g), which is strictly
increasing, bicontinuous, and with (g(-1) > -1 and g(1) < 1) or
(g^-1(-1) > -1 and g^-1(1) < 1). Furthermore g = G U. h, where G is
the identity on [-1/2,-1/3] U [-1/4,-1/5] U ... U [-1/2n,-1(2n+1)] U
[1/(2n+1),1/(2n)] U [1/(2n-1),1/(2n-2)] U ... U [1/3,1/2], n >= 1, and
h is finite and symmetric.
Proof: Will appear in the ms. QED
LEMMA 3.3.12. (ZC) Every g of the form in Lemma 3.3.11 is ME embed usable.
Proof: Write g = G U. h, g::Q]-1,1] into Q[-1,1]. where G is the
identity on [-1/2,-1/3] U [-1/4,-1/5] U ... U [-1/2n,-1(2n+1)] U
[1/(2n+1),1/(2n)] U [1/(2n-1),1/(2n-2)] U ... U [1/3,1/2], n >= 1, and
h is finite, strictly increasing, and symmetric. Let E containedin
Q[0,1]^k and r >= 1. We construct a maximal r-emulation of E that is
f-embedded. Let t = |fld(h)|. We essentially follow the proof of Lemma
3.3.9 except that we need to introduce negative points.
Let D = {0,...,n} x lambda x Q[0,1) and <D be the lexicographic
ordering on D. Let -D be the -x, x in D\{(0,0,0)}, with a purely
formal - that stays in front. Let +-D = D U -D. <+-D is the linear
ordering on +-D which compares elements of D by <D, compares elements
of -D by >D after removing -, and where all of -D is less than all of
D. Let Q[0,1)' be any recursive ordering of Q[0,1) of type omega where
the least element is 0. Let <# be the well ordering of D first by the
ordinal, second by the integer, and third by the rational, where we
use Q[0,1)'. Let <+-# extend <# to D+- by inserting -x right after
every x in D\{(0,0,0)} so that the terms after (0,0,0) alternate
between D and -D. Finally, let <#^k be the well ordering of +-D^k
first by the <+-# of the k coordinates, and then lexicographically by
the <+-# of the k coordinates.
Now let S* containedin +-D^k be the maximal r-emulation of E which is
<+-#^k greedy. Note that S* is definable over (V(lambda),epsilon). Let
kappa_1 < ... < kappa_tn be first order indiscernibles over
V(lambda,epsilon).
We now want to lift symmetric f::Q[-1,1] into Q[-1,1] to symmetric
f*::+-D into +-D in the sense that for x in dom(f*), f(-x) = -f(x). We
lift dom(G) to dom(G)*, G to G*, fld(h) to fld(h)*, h to h*, and
finally f to f*. G is the disjoint union of 2n nondegenerate closed
intervals in Q[-1,1], with n of them in D- and n of them in D. Assign
elements of +-D to these successive endpoints as follows. The left
endpoints get assigned
-(n,0,0),-(n-1,0,0),...,-(1,0,0),(1,0,0),(2,0,0),...,(n,0,0). The
right endpoints get assigned (1,1,0), (2,1,0), ..., (n,1,0). dom(G)*
is the union of the corresponding closed intervals in (+-D,<+-D). G*
is the identity on dom(G)*. Now assign the initial segment of
(i,kappa_0,1+(i-1)t), (i,kappa_1,0), ..., (i,kappa_it) to the elements
of fld(h) between the i-th and (i+1)-th interval of dom(G) intersect
D, 1 <= i < n, in increasing order, and the tail of
(n,kappa_1+(n-1)t,0), ..., (n,kappa_tn,0) to the elements of fld(h)
above the last interval of dom(G). Also assign the initial segment of
-(i,kappa_0,1+(i-1)t), -(i,kappa_1,0), ..., -(i,kappa_it) to the
elements of fld(h) between the i-th and (i+1)-th interval of dom(G)
intersect -D, 1 <= i < n, in increasing order, and the tail of
-(n,kappa_1+(n-1)t,0), ..., -(n,kappa_tn,0) to the elements of fld(h)
below the first (<+-D) interval of dom(G). fld(h)* is the set of all
of these assignments. Now lift h to h*::fld(h)* into fld(h)*
canonically. Clearly fld(h*) = fld(h)*. Finally, define f* = G* U. h*.
Clearly f* is symmetric.
We claim that our maximal r-emulation of E, S* containedin +-D^k, is
f*-embedded. To see this, consider any k-tuple from dom(G*) and the
elements of dom(h*). When we apply f*, we get the same coordinates
from dom(G*) and we get various coordinates from rng(h*). For the
latter, we preserve the sign (membership in -D or in D), the integer
terms (from {-n,...,-1,1,...,n}), keep the rational term 0, but
generally move the ordinal term, which was an indiscernible and
remains an indiscernible. Note that the ordinal term obtained from
applying f* depends only on the ordinal term of the argument, because
f* is symmetric. The relative order of these second terms is preserved
since h is strictly increasing. Hence by the indiscernibility,
membership in S* remains the same.
Now look at the system (+-D,<+-D,S*,f*), We want to go back down to
the rationals and form the system (Q[-1,1],<,S,f), where we have yet
to define S. Note that <+-D is a dense linear ordering with no
endpoints. We let +-D* be the elements of +-D whose ordinal is <
kappa_nt, together with those whose ordinal is kappa_nt and whose
integer is in {-n+1,...,n-1}, together with +-(n,kappa_nt ,0), and
<+-D* be <+-D restricted to +-D*. Note that <+-D* is a dense linear
ordering with endpoints -(n,kappa_nt.0), (n,kappa_nt,0), and is also
the initial segment of <+-# up through -(n,kappa_nt,0). Note that
f*::+-D* into +-D*. Also by the greedy construction of S*, we see that
S* intersect +-D*^k containedin +-D*^k is a maximal r-emulation of E.
We now form (+-D*,<+-D*,S* intersect +-D*^k,f*). Note that fld(f*) is
countable, and "<+-D is dense" and "S* intersect +-D*^k" is a maximal
r-emulation of E that is f*-embedded" are AE assertions. Therefore by
an inductive construction of length omega, we can form the subsystem
(+-D**,<+-D**,S* intersect +-D**^k,f*), f**::+-D** into +-D**, where
+-D** is countable, <+-D** is countable dense with the same endpoints,
and S* intersect +-D**^k containedin +-D**^k remains a maximal
r-emulation of E that is f*-embedded. We now use any increasing
bijection from <+-D** onto Q[-1,1] to form the isomorphic copy
(Q[-1,1],<,S,f). S containedin Q[-1,1]^k will be a maximal r-emulation
of E that is f-embedded since AE sentences are preserved. QED
Proof of MEE/3: In ZC. We have already established the forward
direction. Now let f::Q[0,1] into Q[0,1] be bicontinuous, order
theoretic, strictly increasing, and where
i. There is no 0 < p,q < 1 such that f maps 0 to p and q to 1.
ii. There is no 0 < p,q < 1 such that f maps p to 0 and 1 to q.
Let f extend to a function g as given by Lemma 3.3.8. It suffices to
show that g is ME embed usable. Case iv is immediate and case v
follows from MEE/2. Cases i,ii are by Lemmas 3.2.9 and 3.2.10. For
case iii, since h alters 0 and 1, we have the four possibilities
a. f(0) > 0, f(1) < 1.
b. f(0) > 0, f^-1(1) < 1.
c. f^-1(0) > 0, f(1) < 1.
d. f^-1(0) > 0, f^-1(1) < 1.
Now b,d contradict i,ii, respectively. Hence we have a or c. Now apply
Lemma 3.3.11 and 3.3.12 to obtain that g is ME embed usable. QED
TO BE CONTINUED. we will be proving MEE/4, this time using far more than ZFC.
************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 756th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/23/16 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 10/25/16 1:25PM
730: Consistency of Mathematics/2 11/17/16 9:50PM
731: Large Cardinals and Emulations/26 11/21/16 5:40PM
732: Large Cardinals and Emulations/27 11/28/16 1:31AM
733: Large Cardinals and Emulations/28 12/6/16 1AM
734: Large Cardinals and Emulations/29 12/8/16 2:53PM
735: Philosophical Geometry/13 12/19/16 4:24PM
736: Philosophical Geometry/14 12/20/16 12:43PM
737: Philosophical Geometry/15 12/22/16 3:24PM
738: Philosophical Geometry/16 12/27/16 6:54PM
739: Philosophical Geometry/17 1/2/17 11:50PM
740: Philosophy of Incompleteness/2 1/7/16 8:33AM
741: Philosophy of Incompleteness/3 1/7/16 1:18PM
742: Philosophy of Incompleteness/4 1/8/16 3:45AM
743: Philosophy of Incompleteness/5 1/9/16 2:32PM
744: Philosophy of Incompleteness/6 1/10/16 1/10/16 12:15AM
745: Philosophy of Incompleteness/7 1/11/16 12:40AM
746: Philosophy of Incompleteness/8 1/12/17 3:54PM
747: PA Incompleteness/2 2/3/17 12:07PM
748: Large Cardinals and Emulations/30 2/15/17 2:19AM
749: Large Cardinals and Emulations/31 2/15/17 2:19AM
750: Large Cardinals and Emulations/32 2/15/17 2:20AM
751: Large Cardinals and Emulations/33 2/17/17 12:52AM
752: Emulation Theory for Pure Math/1 3/14/17 12:57AM
753: Emulation Theory for Math Logic 3/10/17 2:17AM
754: Large Cardinals and Emulations/34 12 00:34:34 EST 2017
755: Large Cardinals and Emulations/35 3/12/17 12:33AM
Harvey Friedman
More information about the FOM
mailing list