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In classical philosophy, paradoxes were created by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretans are liars' when uttered by an ancient Greek Cretan was one of the first recorded versions. Contemporary philosophy sometimes employs the same technique to demonstrate that a supposed concept is meaningless or ill-defined.

In classical philosophy, paradoxes were created by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretans are liars' when uttered by an ancient Greek Cretan was one of the first recorded versions. Contemporary philosophy sometimes employs the same technique to demonstrate that a supposed concept is meaningless or ill-defined.

在古典哲学中，<font color="#ff8000"> 悖论Paradoxes</font>是由<font color='ff8800'>自指</font>概念创造出来的，比如全能悖论问是否存在一个如此强大的存在，以至于它可以创造一块无法举起的石头。埃庇米尼得斯悖论，古希腊克里岛人说的“所有克里岛人都是骗子”是有记载的最早版本之一。当代哲学有时使用同样的技巧来证明一个假定的概念是没有意义的或者定义不明确的。

在古典哲学中，<font color="#ff8000"> 悖论Paradoxes</font>是由<font color='ff8800'>自指</font>概念创造出来的，比如<font color='ff8800'>全能悖论</font>问是否存在一个如此强大的存在，以至于它可以创造一块无法举起的石头。<font color='ff8800'>埃庇米尼得斯悖论</font>，古希腊克里岛人说的“所有克里岛人都是骗子”是有记载的最早版本之一。当代哲学有时使用同样的技巧来证明一个假定的概念是没有意义的或者定义不明确的。

In mathematics and computability theory, self-reference (also known as Impredicativity) is the key concept in proving limitations of many systems. Gödel's theorem uses it to show that no formal consistent system of mathematics can ever contain all possible mathematical truths, because it cannot prove some truths about its own structure. The halting problem equivalent, in computation theory, shows that there is always some task that a computer cannot perform, namely reasoning about itself. These proofs relate to a long tradition of mathematical paradoxes such as Russell's paradox and Berry's paradox, and ultimately to classical philosophical paradoxes.

In mathematics and computability theory, self-reference (also known as Impredicativity) is the key concept in proving limitations of many systems. Gödel's theorem uses it to show that no formal consistent system of mathematics can ever contain all possible mathematical truths, because it cannot prove some truths about its own structure. The halting problem equivalent, in computation theory, shows that there is always some task that a computer cannot perform, namely reasoning about itself. These proofs relate to a long tradition of mathematical paradoxes such as Russell's paradox and Berry's paradox, and ultimately to classical philosophical paradoxes.

在数学和可计算性理论中，自我参照(也被称为不确定性)是证明许多系统局限性的关键概念。哥德尔定理用它来表明，没有一个形式上一致的数学系统可以包含所有可能的数学真理，因为它不能证明关于它自己的结构的一些真理。计算理论中的停机问题等价表明，总有一些计算机不能完成的任务，即关于自身的推理。这些证明关系到数学悖论的悠久传统，如罗素悖论和贝瑞悖论，并最终关系到经典哲学悖论。

在数学和可计算性理论中，<font color='ff8800'>自指</font>(也被称为不确定性)是证明许多系统局限性的关键概念。<font color='ff8800'>哥德尔定理Gödel's theorem</font>用它来表明，没有一个形式上一致的数学系统可以包含所有可能的数学真理，因为它不能证明某些关于它自身结构的真理。计算理论中停机问题的等价问题表明，总有一些计算机不能完成的任务，即关于自身的推理。这些证明关系到数学悖论的悠久传统，如<font color='ff8800'>罗素悖论</font>和<font color='ff8800'>贝瑞悖论</font>，并最终关系到经典哲学悖论。

 

In game theory, undefined behaviors can occur where two players must model each other's mental states and behaviors, leading to infinite regress.

In game theory, undefined behaviors can occur where two players must model each other's mental states and behaviors, leading to infinite regress.

In computer programming, self-reference occurs in reflection, where a program can read or modify its own instructions like any other data. Numerous programming languages support reflection to some extent with varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation) in functional programming, where a code structure refers back to itself during computation. 'Taming' self-reference from potentially paradoxical concepts into well-behaved recursions has been one of the great successes of computer science, and is now used routinely in, for example, writing compilers using the 'meta-language' ML. Using a compiler to compile itself is known as bootstrapping. Self-modifying code is possible to write (programs which operate on themselves), both with assembler and with functional languages such as Lisp, but is generally discouraged in real-world programming. Computing hardware makes fundamental use of self-reference in flip-flops, the basic units of digital memory, which convert potentially paradoxical logical self-relations into memory by expanding their terms over time. Thinking in terms of self-reference is a pervasive part of programmer culture, with many programs and acronyms named self-referentially as a form of humor, such as GNU ('Gnu's not Unix') and PINE ('Pine is not Elm'). The GNU Hurd is named for a pair of mutually self-referential acronyms.

In computer programming, self-reference occurs in reflection, where a program can read or modify its own instructions like any other data. Numerous programming languages support reflection to some extent with varying degrees of expressiveness. Additionally, self-reference is seen in recursion (related to the mathematical recurrence relation) in functional programming, where a code structure refers back to itself during computation. 'Taming' self-reference from potentially paradoxical concepts into well-behaved recursions has been one of the great successes of computer science, and is now used routinely in, for example, writing compilers using the 'meta-language' ML. Using a compiler to compile itself is known as bootstrapping. Self-modifying code is possible to write (programs which operate on themselves), both with assembler and with functional languages such as Lisp, but is generally discouraged in real-world programming. Computing hardware makes fundamental use of self-reference in flip-flops, the basic units of digital memory, which convert potentially paradoxical logical self-relations into memory by expanding their terms over time. Thinking in terms of self-reference is a pervasive part of programmer culture, with many programs and acronyms named self-referentially as a form of humor, such as GNU ('Gnu's not Unix') and PINE ('Pine is not Elm'). The GNU Hurd is named for a pair of mutually self-referential acronyms.

在计算机程序设计中，自引用发生在反射中，程序可以像读取或修改其他数据一样读取或修改自己的指令。许多编程语言在某种程度上支持反射，具有不同程度的表达能力。此外，在函数式编程中的递归(与数学递回关系式相关)中可以看到自引用，其中代码结构在计算过程中反向引用自身。“驯服”自我参照从潜在的矛盾概念到良好的行为递归一直是计算机科学的伟大成功之一，现在经常被用于，例如，编译器使用“元语言”机器学习。使用编译器编译自身称为引导。无论是使用汇编语言还是使用诸如 Lisp 之类的函数式语言，编写程序自修改程序都是可能的，但在实际编程中通常不提倡这样做。计算硬件在触发器中基本利用了自参照，这是数字存储器的基本单位，它通过随时间推移扩展其术语，将潜在的矛盾逻辑自我关系转化为存储器。自我引用思维是程序员文化中一个普遍存在的部分，许多程序和首字母缩略词都是以自我引用的方式命名的，这是一种幽默的形式，比如 GNU (“ GNU 不是 Unix”)和 PINE (“ PINE 不是 Elm”)。GNU Hurd 是以一对相互自我参照的缩写命名的。

在计算机程序设计中，自指发生在反馈中，程序可以像读取或修改其他数据一样读取或修改自己的指令。许多编程语言在某种程度上支持反射，具有不同程度的表达能力。此外，在函数式编程中的递归(与数学递回关系式相关)中可以看到自引用，其中代码结构在计算过程中反向引用自身。“驯服”自我参照从潜在的矛盾概念到良好的行为递归一直是计算机科学的伟大成功之一，现在经常被用于，例如，编译器使用“元语言”机器学习。使用编译器编译自身称为引导。无论是使用汇编语言还是使用诸如 Lisp 之类的函数式语言，编写程序自修改程序都是可能的，但在实际编程中通常不提倡这样做。计算硬件在触发器中基本利用了自参照，这是数字存储器的基本单位，它通过随时间推移扩展其术语，将潜在的矛盾逻辑自我关系转化为存储器。自我引用思维是程序员文化中一个普遍存在的部分，许多程序和首字母缩略词都是以自我引用的方式命名的，这是一种幽默的形式，比如 GNU (“ GNU 不是 Unix”)和 PINE (“ PINE 不是 Elm”)。GNU Hurd 是以一对相互自我参照的缩写命名的。

 

[[Tupper's self-referential formula]] is a mathematical curiosity which plots an image of its own formula.

[[Tupper's self-referential formula]] is a mathematical curiosity which plots an image of its own formula.

Tupper's self-referential formula is a mathematical curiosity which plots an image of its own formula.

Tupper's self-referential formula is a mathematical curiosity which plots an image of its own formula.

 

 

== In biology ==

== In biology ==