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By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following fact.

Theorem (CH). Under CH the real line contains an uncountable subset $X$ admitting no strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z$ of $X$.

On the other hand, a known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem (PFA). Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z\subset X$.

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)?

By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following fact.

Theorem (CH). Under CH the real line contains an uncountable subset $X$ admitting no strictly decreasing function $f:Z\to X$, defined on some uncountable subset $Z$ of $X$.

On the other hand, a known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem (PFA). Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on some uncountable subset $Z\subset X$.

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)?

A known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem. Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z\subset X$.

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)? What happens with this theorem under CH or $\Diamond$?

By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following fact.

Theorem (CH). Under CH the real line contains an uncountable subset $X$ admitting no strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z$ of $X$.

On the other hand, a known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem (PFA). Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z\subset X$.

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)?

The following theorem can be proved by the standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line):

Theorem (CH). Under CH there exits an uncountable subset $X$ of the real line admitting no strictly decreasing function $f:Z\to X$ defined on an uncountable subset $Z\subset X$.

On the other hand, a known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem (PFA). Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z\subset X$.

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)?

A known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem. Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on an uncountable subset $Z\subset X$.

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)? What happens with this theorem under CH or $\Diamond$?